Abstract and Applied Analysis

VolumeΒ 2011, Article IDΒ 582576, 7 pages

http://dx.doi.org/10.1155/2011/582576

## On -Convergence in Measure of a Sequence of Measurable Functions

^{1}Department of Mathematics and Computer Sciences, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy^{2}Department of Mathematics, University of Athens, Panepistimiopolis, 15784 Athens, Greece

Received 18 November 2010; Accepted 22 March 2011

Academic Editor: AgacikΒ Zafer

Copyright Β© 2011 A. Boccuto et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the study by Papanastassiou and Papachristodoulos, 2009 the notion of -convergence in measure was introduced. In a natural way -convergence in measure induces an equivalence relation on the space of all sequences of measurable functions converging in measure to zero. We show that the quotient space is a complete but not compact metric space.

#### 1. Introduction

Convergence in measure plays a fundamental role in several branches of Mathematics, for example in integration theory and in stochastic processes. In [1] a βBochner-typeβ integration theory was developed in the context of Riesz spaces with respect to a convergence introduced axiomatically, and in particular some Vitali convergence theorems and Lebesgue dominated convergence theorems were proved. Similar subjects were investigated by HaluΕ‘ka and HutnΓk [2, 3] in the setting of operator theory for Bochner- and Dobrakov-type integrals (see also [4, 5]) and in [6β8] for the Kurzweil-Henstock integral in Riesz spaces.

In several contexts of integration theory it could be advisable to extend the concept of convergence in measure in order to get applications, for example, in the study of the stochastic integral and stochastic differential equations (see, e.g., [9]).

This paper is a continuation of [10], where the notion of -convergence in measure was introduced. In this paper we investigate a structure related to the vector space of all converging sequences of measurable functions.

Let be an arbitrary measure space, where is a -valued measure, and let , be measurable functions.

We adopt the following usual terminology. By the notation we denote that the sequence of measurable functions converges in measure to . Also for a pair and we set We denote by the set of all positive integers and the set of all real-valued nonnegative sequences converging to 0. Also for we set Convergence in measure is characterized by elements of as follows: Taking into account that the sequence above expresses the quality of approximation of to , in [10] the authors introduced the following notion of convergence which we call -convergence in measure.

More precisely we say that, given , *-converges in measure to * (and we write ) if and only if there exists an element such that
Obviously -convergence in measure implies convergence in measure. It is proved (see [10, Prepositionβ2.3]) that if the measure is not trivial and , then -convergence in measure implies -convergence in measure, while the converse implication in general fails. So -convergence in measure is strictly stronger than convergence in measure. As a consequence of the above result we have that
where
We note that is considered as a vector space under usual operations and the notation means that is a proper vector space of .

#### 2. Metric Spaces of Sequences of Measurable Functions

In a natural way convergence in measure induces an equivalence relation on the vector space . We consider as a subspace of , copies of the vector space of all real-valued measurable functions with the usual operations.

*Definition 2.1. *Let be elements of . We say that , are equivalent if and only if for each positive real number there exists an element of such that
or equivalently

Since is a vector subspace of the relation *~* is an equivalence one. We set .

In the sequel we will define a metric on under which turns to be a complete metric space, similarly as a FrΓ©chet space.

*Definition 2.2. *Let . We define
where

By (1.5) it follows that is an interval in .

*Remarks 2.3. *(i) We note that the above set could be empty. In this case we set .

(ii) If , then . (Indeed, as and , it follows that if and only if .)

(iii) We set , and hence is a proper vector subspace of and the following strict inclusion holds:

Proposition 2.4. *The function satisfies the following properties: *(i)(ii)* iff *(iii)(iv)*.*

Hence, becomes a metric space and the metric is invariant under translations.

*Proof. *(i) It is obvious.

(ii) If , then .

Conversely, if , then, for each , and hence and consequently .

(iii) For , it holds that
for each sequence of positive real numbers. Hence,
which means that .

(iv) The inequality is obvious if or .

Suppose . Then we conclude that . Hence, implies . So , which implies that

Theorem 2.5. *The space is a complete metric space. *

*Proof. *Let be a Cauchy sequence in , where . Hence, there exists an increasing sequence of positive integers such that
This means that, for each , there exists a sequence of positive real numbers with such that
Then,
From (2.10) and (2.11), proceeding by induction, it follows that there exists an increasing sequence of positive integers such that for each we have
which express the uniform convergence to zero of finite number of sequence which converges to zero and a finite number of tails of convergence series.

We set
By (2.14) it follows that .

We have to show that
This means that we have to find and for a sequence of positive real numbers with such that
Indeed let , and for some .

We set
and so on.

It holds that
Also, for , we have that
(by definition of ). Hence, by (2.12), we take
This implies (2.17), and the proof is complete.

*Remarks 2.6. *It is easy to see the following (a)The addition is continuous.(b)The translation is a homeomorphism. Hence, the system of neighborhoods of determines the topology of .(c)The multiplication operator
is a homeomorphism.(d)The multiplication is not continuous. (If , and , , and for , it holds that
but ).(e)The family is a system of neighborhoods of . (Indeed, if for , then for we have .)

Though is not a topological vector space, is complete and the subspaces , constitute a system of closed and convex neighborhoods of , as we will see in the sequel (Proposition 2.7). Hence, is something like a FrΓ©chet space. For example, the principle of uniform boundedness holds true, as for this principle only continuity of is needed (see [11]).

Proposition 2.7. *The subspaces are closed for each . *

*Proof. *Suppose that and is a sequence in , where , , and such that
Hence, there exist and such that
This implies that , and, since , it follows that .

Proposition 2.8. * is a closed subspace of .*

*Proof. *Suppose , then is a neighborhood of and for all .

Indeed, if for some and some , then , where , which is a contradiction. Hence, , which implies that is closed.

*Remark 2.9. *If denotes the open sphere with center and radius , it is easy to see that the family
is an open covering of without a finite subcovering. Hence is not compact.

#### Acknowledgment

N. Papanastassiou work is supported by the Universities of Perugia and Athens.

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