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On conditional mean ergodic semigroups of random linear operators

Xia Zhang

Author Affiliations

Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin, 300160, People’s Republic of China

Journal of Inequalities and Applications 2012, 2012:150  doi:10.1186/1029-242X-2012-150


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/150


Received:7 February 2012
Accepted:8 May 2012
Published:3 July 2012

© 2012 Zhang; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, we prove two forms of conditional mean ergodic theorem for a strongly continuous semigroup of random isometric linear operators generated by a semigroup of measure-preserving measurable isomorphisms, one of which generalizes and improves several known important results.

1 Introduction and the main results

The notion of a random normed module (briefly, an RN module), which was first introduced in [1] and subsequently elaborated in [2], is a random generalization of that of a normed space. In the last 10 years, the theory of RN modules together with their random conjugate spaces have undergone a systematic and deep development [3-9], in particular the random reflexivity based on the theory of random conjugate spaces and the study of semigroups of random linear operators have also obtained some substantial advances in [6,8,10-12].

It is well known that the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology induced by the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M2">View MathML</a>-norm on an RN module is exactly the topology of convergence in probability P. Actually, it is Mustari and Taylor that earlier observed the essence of the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology, studied probability theory in Banach spaces and did many excellent works [13,14] under the framework of the special RN module <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4">View MathML</a> is the RN module of equivalence classes of X-valued random variables defined on a probability space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6">View MathML</a>, see [4] or Section 2 for the construction of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4">View MathML</a>. Motivated by these works, we have recently begun to study the mean ergodic theorem under the framework of RN modules to obtain the mean ergodic theorem in the sense of convergence in probability, in particular we proved a mean ergodic theorem for a strongly continuous semigroup of random unitary operators defined on complete random inner product modules in [8] and further investigated the mean ergodicity for an almost surely bounded strongly continuous semigroup of random linear operators on a random reflexive RN module in [11]. Based on these and motivated by the idea of [15,16], the purpose of this article is to investigate the conditional mean ergodicity for a special semigroup of random linear operator on the RN module <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8">View MathML</a> and the construction of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8">View MathML</a> is detailed as follows.

Now let us first recall the construction of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M10">View MathML</a> in [16]. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M11">View MathML</a> be a probability space, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M12">View MathML</a> a sub σ-algebra of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M13">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M14">View MathML</a> (or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M15">View MathML</a>) the set of equivalence classes of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M16">View MathML</a>-measurable extended real-valued (real-valued) random variables on Ω. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M17">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M18">View MathML</a>. Similarly, one can understand such notions as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M19">View MathML</a><a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M20">View MathML</a><a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M21">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M22">View MathML</a>. Define the mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M23">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M24">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M25">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M26">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M27">View MathML</a> denotes the extended conditional expectation, and let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M28">View MathML</a>

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M29">View MathML</a> is an RN module. In fact, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M10">View MathML</a> is exactly the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M20">View MathML</a>-module generated by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M32">View MathML</a>, namely <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M33">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M34">View MathML</a>. Further, motivated by the idea of constructing <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M10">View MathML</a>, Guo in [3] constructed a more general RN module <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M36">View MathML</a> and established the random conjugate representation theorem of this type of RN module. Precisely speaking, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37">View MathML</a> be an RN module over the scalar field K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M11">View MathML</a>, the mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M39">View MathML</a> is defined as follows for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M41">View MathML</a>:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M42">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M43">View MathML</a> also denotes the extended conditional expectation and let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M44">View MathML</a>

Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M45">View MathML</a> is an RN module over K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M46">View MathML</a>. If we take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M47">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M36">View MathML</a> is exactly <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M10">View MathML</a>; if we further take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M50">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M51">View MathML</a> (briefly, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8">View MathML</a>) is an RN module over K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M46">View MathML</a>.

We can now state the main results of this article as follows.

Theorem 1.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M54">View MathML</a>be a probability space, Xa reflexive Banach space overK, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M55">View MathML</a>a semigroup of measure-preserving measurable isomorphisms on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M11">View MathML</a>, pan arbitrary positive number such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M57">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M58">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M59">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M60">View MathML</a>, then, for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61">View MathML</a>, there exists some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M62">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M63">View MathML</a>for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64">View MathML</a>and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M65">View MathML</a>

(1)

in the<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology induced by<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M67">View MathML</a>.

Theorem 1.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M54">View MathML</a>, X, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M55">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M12">View MathML</a>be the same as in Theorem 1.1 and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M71">View MathML</a>a given positive number such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M72">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M73">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M74">View MathML</a>. Then, for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M75">View MathML</a>, there exists some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M76">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M63">View MathML</a>for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64">View MathML</a>and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M79">View MathML</a>

(2)

in the<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology induced by<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M81">View MathML</a>. In particular, if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M82">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M83">View MathML</a>.

The remainder of this article is organized as follows: in Section 2 we briefly recall some necessary basic notions and facts and Section 3 is devoted to the proof of our main results.

2 Preliminaries

In the sequel of this article, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6">View MathML</a> denotes a probability space, K the scalar field R of real numbers or C of complex numbers, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M85">View MathML</a> the algebra of equivalence classes of K-valued <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M13">View MathML</a>-measurable random variables on Ω under the ordinary addition, scalar multiplication and multiplication operations on equivalence classes. Besides, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M15">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M14">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M89">View MathML</a> be the same as in Section 1.

It is well known from [17] that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M90">View MathML</a> is a complete lattice under the ordering ≤: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M91">View MathML</a> if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M92">View MathML</a> for P-almost all ω in Ω (briefly, a.s.), where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M93">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M94">View MathML</a> are arbitrarily chosen representatives of ξ and η, respectively. Furthermore, every subset A of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M90">View MathML</a> has a supremum, denoted by ⋁A, and an infimum, denoted by ⋀A, and there exist two sequences <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M96">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M97">View MathML</a> in A such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M98">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M99">View MathML</a>. Finally, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M100">View MathML</a>, as a sublattice of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M90">View MathML</a>, is complete in the sense that every subset with an upper bound has a supremum.

Definition 2.1[2,3]

An ordered pair <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37">View MathML</a> is called an RN module over K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6">View MathML</a> if S is a left module over the algebra <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M85">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M105">View MathML</a> is a mapping from S to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M89">View MathML</a> such that the following three axioms are satisfied:

(1) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M107">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M108">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40">View MathML</a>;

(2) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M110">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M111">View MathML</a>;

(3) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M112">View MathML</a> implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M113">View MathML</a> (the null vector of S), where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M105">View MathML</a> is called the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M2">View MathML</a>-norm on S and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M116">View MathML</a> is called the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M2">View MathML</a>-norm of a vector <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40">View MathML</a>.

It should be pointed out that the following idea of introducing the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology is due to Schweizer and Sklar [18].

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37">View MathML</a> be an RN module over K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6">View MathML</a>. For any positive real numbers ε and λ such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M122">View MathML</a>, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M123">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M124">View MathML</a> is a local base at the null vector θ of some Hausdorff linear topology. The linear topology is called the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology. In this article, given an RN module <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37">View MathML</a> over K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6">View MathML</a>, it is always assumed that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37">View MathML</a> is endowed with the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology. One only needs to notice that a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M130">View MathML</a> in S converges to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40">View MathML</a> in the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M133">View MathML</a> converges to 0 in probability P.

Example Let X be a normed space over K and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4">View MathML</a> the linear space of equivalence classes of X-valued <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M13">View MathML</a>-random variables on Ω. The module multiplication operation <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M136">View MathML</a> is defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M137">View MathML</a> the equivalence class of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M138">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M93">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M140">View MathML</a> are the respective arbitrarily chosen representatives of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M141">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M142">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M143">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M144">View MathML</a>. Furthermore, the mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M145">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M146">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M147">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M140">View MathML</a> is as above. Then it is easy to see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M149">View MathML</a> is an RN module over K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M150">View MathML</a>.

Definition 2.2[19]

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M151">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M152">View MathML</a> be two RN modules over K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6">View MathML</a>. A linear operator T from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M154">View MathML</a> to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M155">View MathML</a> is called a random linear operator, further, the random linear operator T is called a.s. bounded if there exists some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M156">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M157">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M158">View MathML</a>. Denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M159">View MathML</a> the linear space of a.s. bounded random linear operators from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M154">View MathML</a> to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M155">View MathML</a>, define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M162">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M163">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M164">View MathML</a>, then it is easy to see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M165">View MathML</a> is an RN module over K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6">View MathML</a>.

Specially, denote <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M167">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M168">View MathML</a> when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M151">View MathML</a> is a given RN module <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37">View MathML</a> over K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M171">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M172">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M168">View MathML</a> is called the random conjugate space of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M175">View MathML</a> be the random conjugate space of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M168">View MathML</a>. The canonical embedding mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M177">View MathML</a> defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M178">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M180">View MathML</a>, is random-norm preserving. If J is surjective, then S is called random reflexive [10].

Definition 2.3[8,11]

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37">View MathML</a> be an RN module over K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M150">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M183">View MathML</a> the set of a.s. bounded random linear operators on S. A family <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M184">View MathML</a> is called a semigroup of random linear operators if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M185">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M186">View MathML</a>, where I denotes the identity operator on S. Further, if the mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M187">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M188">View MathML</a> is continuous w.r.t. the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40">View MathML</a>, then the semigroup of random linear operators <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M191">View MathML</a> is said to be strongly continuous. Besides, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M192">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M191">View MathML</a> is called an a.s. bounded strongly continuous semigroup of random linear operators.

Proposition 2.4[19]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M151">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M152">View MathML</a>be twoRNmodules overKwith base<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6">View MathML</a>. Then we have the following statements:

(1) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M164">View MathML</a>if and only ifTis a continuous module homomorphism;

(2) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M164">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M199">View MathML</a>, where 1 denotes the identity element in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M15">View MathML</a>.

3 Proof of the main results

The proof of Theorem 1.2 needs Theorem 1.1 and Lemma 3.5 below. To prove Theorem 1.1 and introduce Lemma 3.5, we will first recall the definition of Riemann integral for abstract-valued functions from a finite real interval to an RN module and a sufficient condition for such a function to be Riemann-integrable.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201">View MathML</a> be a finite real interval and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M202">View MathML</a> a finite partition into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201">View MathML</a>, namely, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M204">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M205">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M206">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M207">View MathML</a>). Besides, in the following of this section we always suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37">View MathML</a> denotes a complete RN module over K with base <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M150">View MathML</a>.

Definition 3.1[8]

Let f be a function from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201">View MathML</a> to S. f is called Riemann-integrable on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201">View MathML</a> if there exists some I in S with the following property: for any positive numbers ε and λ with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M122">View MathML</a> there is a positive number <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M213">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M214">View MathML</a>

for any finite partition <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M215">View MathML</a> and arbitrarily chosen <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M216">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M217">View MathML</a>) whenever <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M218">View MathML</a>. Further I is called the Riemann integral of f in the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology over <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201">View MathML</a>, denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M221">View MathML</a>.

Proposition 3.2[8]

Letfbe a continuous function from<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201">View MathML</a>toSsuch that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M223">View MathML</a>, thenfis Riemann integrable in the<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201">View MathML</a>. Further, if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M226">View MathML</a>, thenfis Riemann integrable in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M227">View MathML</a>on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M227">View MathML</a>denotes the 2-norm of the Banach space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M230">View MathML</a>.

Definition 3.3[11]

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M191">View MathML</a> be an a.s. bounded strongly continuous semigroup of random linear operators on an RN module S. We denote by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M232">View MathML</a>

the Cesàro means of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M191">View MathML</a>. For any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40">View MathML</a>, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M235">View MathML</a> converges to some point in S as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M236">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M191">View MathML</a> is called mean ergodic.

Note that it is Proposition 3.2 that makes Definition 3.3 be well defined.

Proposition 3.4[11]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M37">View MathML</a>be a completeRNmodule overKwith base<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M6">View MathML</a>. IfSis random reflexive, then every a.s. bounded strongly continuous semigroup of random linear operators onSis mean ergodic.

It is known from [9] that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4">View MathML</a> is random reflexive if and only if X is a reflexive Banach space. Besides, Guo [3] proved that if an RN module S is random reflexive, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M36">View MathML</a> is also random reflexive. Based on these facts as well as the preceding Proposition 3.4, we can now prove Theorem 1.1 as follows.

Proof of Theorem 1.1 For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64">View MathML</a>, define the mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M243">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M244">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M245">View MathML</a>, then it is clear that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M246">View MathML</a> is a module homomorphism. Furthermore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M247">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M249">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M250">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61">View MathML</a>, i.e. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M252">View MathML</a> is a semigroup of random linear operators on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8">View MathML</a>.

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M254">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M257">View MathML</a>, it follows that

(3)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M259">View MathML</a>. Thus we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M260">View MathML</a>

(4)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M257">View MathML</a>, letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M264">View MathML</a> in (4) yields that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M265">View MathML</a>

namely <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M266">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61">View MathML</a>, which shows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M268">View MathML</a>

is a random isometric linear operator for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64">View MathML</a>. Moreover,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M270">View MathML</a>

Thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M252">View MathML</a> is a strongly continuous semigroup of random isometric linear operators on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8">View MathML</a>.

Clearly, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M252">View MathML</a> is an a.s. bounded strongly continuous semigroup of random linear operators on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M274">View MathML</a>. Since X is reflexive, it follows that the RN module <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M4">View MathML</a> is random reflexive, further, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M8">View MathML</a> is random reflexive. Thus it follows from Proposition 3.4 that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M246">View MathML</a> is mean ergodic for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64">View MathML</a>, i.e. for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M61">View MathML</a> there exists some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M62">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M281">View MathML</a>

Now it remains to show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M63">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64">View MathML</a>. In fact, since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M284">View MathML</a>

(5)

and

(6)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M286">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M40">View MathML</a>, fix s and x, letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M236">View MathML</a> in (5) yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M289">View MathML</a>

Further, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M246">View MathML</a> is a random isometric linear operator for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M291">View MathML</a>, thus we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M292">View MathML</a>, namely <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M63">View MathML</a>.

This completes the proof. □

It should be pointed out that Lemma 3.5 is merely mentioned in [8] and partially proved in [12].

Lemma 3.5Letfbe a continuous function from<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201">View MathML</a>to<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M15">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M296">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M297">View MathML</a>

(7)

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M298">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M156">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M300">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M301">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M302">View MathML</a> is a sequence of pairwise disjoint <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M303">View MathML</a>-measurable sets such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M304">View MathML</a>. Define a mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M305">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M306">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M301">View MathML</a>. Obviously, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M308">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M301">View MathML</a> and each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M310">View MathML</a>, thus it follows by Proposition 3.2 that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M311">View MathML</a> is Riemann integrable in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M227">View MathML</a> on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M201">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M314">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M315">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M316">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M317">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M318">View MathML</a>) be the same as in Definition 3.1 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M319">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M320">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M301">View MathML</a>. Since

(8)

for each fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M301">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M324">View MathML</a>

Thus for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M325">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M326">View MathML</a>

(9)

Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M327">View MathML</a>

it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M328">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M329">View MathML</a> are both Cauchy sequences. Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M330">View MathML</a> in Equation (9) yields Equation (7). □

We can now prove Theorem 1.2.

Proof of Theorem 1.2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M244">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M333">View MathML</a>, then it follows from Theorem 1.1 that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M334">View MathML</a>

in the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology induced by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M336">View MathML</a>. Thus by the random Schwarz-Cauchy inequality, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M337">View MathML</a>

in probability P. For any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M75">View MathML</a>, since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M339">View MathML</a>

for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M257">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M341">View MathML</a> is dense in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M342">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M81">View MathML</a>, one can obtain the desired Equation (2).

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M82">View MathML</a>, it suffices to show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M345">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M259">View MathML</a>, then for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M257">View MathML</a>, it follows from Lemma 3.5 that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M348">View MathML</a>

(10)

Furthermore, since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M349">View MathML</a>

in the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M1">View MathML</a>-topology induced by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M81">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M352">View MathML</a>

for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M353">View MathML</a>, letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M354">View MathML</a> in (10), it follows from the Lebesgue’s dominated convergence theorem that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M355">View MathML</a>

namely the desired result follows. □

Let H be a Hilbert space over K. It is clear that Theorem 1.2 still holds when X is taken place of H, which includes the following known result.

Corollary 3.6[12]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M356">View MathML</a>be a probability space, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M55">View MathML</a>a semigroup of measure-preserving measurable isomorphisms on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M54">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M359">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M360">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M361">View MathML</a>. Then, for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M362">View MathML</a>, there exists some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M363">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M63">View MathML</a>for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M64">View MathML</a>and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M366">View MathML</a>

(11)

Corollary 3.7If, in addition to the hypothesis of Theorem 1.2, we assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M259">View MathML</a>if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M368">View MathML</a>of 1, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/150/mathml/M369">View MathML</a>.

Competing interests

The author declare that they have no competing interests.

Acknowledgement

The author would like to express his sincere gratitude to Prof. Guo Tiexin for his invaluable suggestions. The study was supported by the National Natural Science Foundation of China (No. 11171015). The author would also like to thank the referees for their invaluable suggestions.

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