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

Authors: A. Boccuto; Ch. Papachristodoulos; N. Papanastassiou
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

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


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## Body

## 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 “Bochner-type” 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 Haluška and Hutník [2, 3] in the setting of operator theory for Bochner- and Dobrakov-type integrals (see also [4, 5]) and in [6–8] 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 p-convergence in measure was introduced. In this paper we investigate a structure related to the vector space M of all converging sequences of measurable functions.Let(Γ,Σ,μ) be an arbitrary measure space, where μ is a [0,∞]-valued measure, and let fn,f:Γ→ℝ,n=1,2,…, be measurable functions.We adopt the following usual terminology. By the notationfn→μf we denote that the sequence of measurable functions (fn)n converges in measure to f. Also for a pair ((fn)n,f) and ε≥0 we set (1.1)Anε={γ∈Γ:|fn(γ)-f(γ)|≥ε}={|fn-f|≥ε},n=1,2,….
We denote by ℕ the set of all positive integers and c0+ the set of all real-valued nonnegative sequences (εn)n converging to 0. Also for p>0 we set (1.2)lp+={(εn)n:εn≥0forn=1,2,…,∑n=1∞εnp<∞}.
Convergence in measure is characterized by elements (εn)n of c0+ as follows: (1.3)fn⟶μfiffthereexists(εn)n∈c0+such thatlim⁡n→∞μ(Anεn)=0.
Taking into account that the sequence (εn)n above expresses the quality of approximation of (fn)n to f, in [10] the authors introduced the following notion of convergence which we call p-convergence in measure.More precisely we say that, givenp>0, (fn)np-converges in measure tof  (and we write fn→p-μf) if and only if there exists an element (εn)n∈ℓp+ such that (1.4)lim⁡n→∞⁡μ(Anεn)=0.
Obviously p-convergence in measure implies convergence in measure. It is proved (see [10, Preposition 2.3]) that if the measure μ is not trivial and 0<p<q, then p-convergence in measure implies q-convergence in measure, while the converse implication in general fails. So p-convergence in measure is strictly stronger than convergence in measure. As a consequence of the above result we have that (1.5)M0≨Mp≨Mq≨M∞≨M,0<p<q,
where(1.6)M={(fn)n:fn⟶μ0},Mp={(fn)n:fn→p-μ0},p>0,M0=⋂p>0Mp,M∞=⋃p>0Mp.
We note that M is considered as a vector space under usual operations and the notation N≨M means that N is a proper vector space of M.


## 2. Metric Spaces of Sequences of Measurable Functions

In a natural wayp-convergence in measure induces an equivalence relation on the vector space M={(fn)n:fn→μ0}. We consider M as a subspace of L0(Γ)ℕ, ℵ0 copies of the vector space L0(Γ) of all real-valued measurable functions with the usual operations.Definition 2.1.
Let(fn)n,(gn)n be elements of M. We say that (fn)n, (gn)n are equivalent ((fn)n~(gn)n) if and only if for each positive real number p there exists an element (εn)n of ℓp+ such that
(2.1)lim⁡n→∞μ([|fn-gn|>εn])=0
or equivalently
(2.2)fn-gn→p-μ0,∀p>0⟺(fn-gn)n∈M0.
SinceM0 is a vector subspace of M the relation ~ is an equivalence one. We set ℳ=M~=M/M0.
In the sequel we will define a metricd on ℳ under which ℳ turns to be a complete metric space, similarly as a Fréchet space.Definition 2.2.
Let(fn)n∈ℳ. We define
(2.3)‖(fn)n‖=arctan(inf⁡A(fn)),
where
(2.4)A(fn)={p>0:fn→p-μ0}={p>0:(fn)n∈Mp}.
By (1.5) it follows that A(fn) is an interval in ℝ.Remarks 2.3.
(i) We note that the above setA(fn) could be empty. In this case we set ∥(fn)n∥=π/2.
(ii) If(fn)n~(gn)n, then ∥(fn)n∥=∥(gn)n∥. (Indeed, as (gn)n=(gn-fn)n+(fn)n and (fn)n∈Mp, it follows that (fn)n∈Mp if and only if (gn)n∈Mp.)
(iii) We setℳp=Mp/~,p>0, and hence ℳp is a proper vector subspace of ℳ and the following strict inclusion holds:
(2.5)Mp1≨Mp2if0<p1<p2(see(1.5)).Proposition 2.4.
The function∥∥:ℳ→ℝ satisfies the following properties: (i)
∥(fn)n∥≥0(ii)
∥(fn)n∥=0 iff (fn)n~(0)n(iii)
∥a(fn)n∥=∥(fn)n∥fora≠0(iv)
∥(fn+gn)n∥≤∥(fn)n∥+∥(gn)n∥.Hence,ℳ becomes a metric space and the metric d((fn)n,(gn)n)=∥(fn-gn)n∥ is invariant under translations.Proof.
(i) It is obvious.
(ii) If(fn)n~(0)n, then ∥(fn)n∥=0.
Conversely, if∥(fn)n∥=0, then, (fn)n→p-μ0 for each p>0, and hence (fn)n∈ℳ0 and consequently (fn)~(0)n.
(iii) Fora≠0, it holds that
(2.6)Anεn=[|fn|≥εn]=[|afn|≥|a|εn],∑n=1∞εnp<∞⟺∑n=1∞(|a|εn)p<∞,
for each sequence (εn)n of positive real numbers. Hence,
(2.7)(fn)n→p-μ0iff(afn)n→p-μ0,
which means that ∥(fn)n∥=∥(afn)n∥.
(iv) The inequality is obvious if∥(fn)n∥=π/2 or ∥(gn)n∥=π/2.
Suppose∥(fn)n∥≤∥(gn)n∥<π/2. Then we conclude that A(gn)⊂A(fn). Hence, (gn)n→p-μ0 implies (fn)n+(gn)n→p-μ0. So A(gn)⊆A(fn+gn), which implies that
(2.8)‖(fn+gn)n‖≤‖(gn)n‖≤‖(fn)n‖+‖(gn)n‖.Theorem 2.5.
The space(ℳ,d) is a complete metric space.Proof.
Let(Fn)n be a Cauchy sequence in ℳ, where Fn=(fn,i)i,n=1,2,…. Hence, there exists an increasing sequence of positive integers (nk)k such that
(2.9)‖Fn-Fm‖<arctan1k,forn,m≥nk,k=1,2,….
This means that, for each n,m≥nk, there exists a sequence (εn,m,i)i of positive real numbers with ∑i=1∞εn,m,i1/k<∞ such that
(2.10)μ(An,m,i)⟶0,i⟶∞,whereAn,m,i=[|fn,i-fm,i|≥εn,m,i].
Then,
(2.11)∑i=1∞(max⁡nl≤n≤nk+1⁡εn,nk+1,i)1/l<∞,forl=1,2,…,k,k∈N.
From (2.10) and (2.11), proceeding by induction, it follows that there exists an increasing sequence (ik)k of positive integers such that for each k we have
(2.12)μ(An,nk+1,i)<1kfori≥ik,n1≤n≤nk+1,k=1,2,…,(2.13)∑i=ik∞(max⁡nl≤n≤nk+1⁡εn,m,i)1/l<12k,forl=1,2,…,k,(2.14)μ[|fnk+1,i|≥1k]<1k,fori≥ik,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(2.15)F=(fn1,1,fn1,2,…,fn1,i1-1;fn2,i1,…,fn2,i2-1;…;fnk+1,ik,…,fnk+1,ik+1-1;…)=(fi)i.
By (2.14) it follows that F=(fi)i∈ℳ.
We have to show that(2.16)‖Fn-F‖⟶0,n⟶∞⟺∀l∈N∃n0∈N:(fn,i-fi)i⟶(1/l)-μ0,forn≥n0.
This means that we have to find n0∈ℕ and for n≥n0 a sequence of positive real numbers (εi)i with ∑i=1∞εi1/ℓ<∞ such that
(2.17)μ(Ai)⟶0,i⟶∞,whereAi=[|fn,i-fi|≥εi].
Indeed let ℓ∈ℕ,n≥n0=nℓ, and nℓ≤nk<n<nk+1 for some k∈ℕ.
We set(2.18)εi=1,ifi=1,2,…,ik-1,εi=εn,nk+1,i,ifi=ik,…,ik+1-1,εi=εn,nk+2,i,ifi=ik+1,…,ik+2-1,
and so on.
It holds that(2.19)∑i=1∞εi1/l≤(ik-1)+12k+12k+1+⋯<∞(by(2.13)).
Also, for im≤i≤im+1,m≥k, we have that
(2.20)Ai=[|fn,i-fi|≥εi]=[|fn,i-fnm+1,i|≥εn,nm+1,i]
(by definition of fi). Hence, by (2.12), we take
(2.21)μ(Ai)<1m.
This implies (2.17), and the proof is complete.Remarks 2.6.
It is easy to see the following(a)
The addition+:((fn)n,(gn)n)↦(fn+gn)n is continuous.(b)
The translationT(gn)n:(fn)n↦(fn)n+(gn)n=(fn+gn)n is a homeomorphism. Hence, the system of neighborhoods of (0)n determines the topology of (ℳ,d).(c)
The multiplication operator(2.22)Ha:(fn)n⟼a⋅(fn)n,a≠0,
is a homeomorphism.(d)
The multiplication(a,(fn)n)↦a(fn)n is not continuous. (If an→0,an≠0,n=1,2,…, and F=(fn)n∈ℳ, F≠0, and Fn=F for n=1,2,…, it holds that
(2.23)an⟶a≠0,Fn⟶dF,
but ∥anFn-0F∥=∥anFn∥=∥Fn∥=∥F∥↛0).(e)
The family(ℳp)p>0 is a system of neighborhoods of (0)n. (Indeed, if Sr=S((0)n,r)={(fn)n:∥(fn)n∥<r} for r>0, then for 0<r1<p<r2 we have Sr1⊂ℳp⊂Sr2.)
Though(ℳ,d) is not a topological vector space, (ℳ,d) is complete and the subspaces ℳp, p>0 constitute a system of closed and convex neighborhoods of (0)n, as we will see in the sequel (Proposition 2.7). Hence, (ℳ,d) is something like a Fréchet space. For example, the principle of uniform boundedness holds true, as for this principle only continuity of Ha is needed (see [11]).Proposition 2.7.
The subspacesℳp are closed for each  p>0.Proof.
Suppose thatp>0 and (Fn)n is a sequence in ℳp, where Fn=(fn,i)i, n=1,2,…, and F=(fn)n∈ℳ such that
(2.24)Fn⟶dF⟺‖Fn-F‖⟶0,n→∞.
Hence, there exist p′<p and n0 such that
(2.25)Fn0-F→p′-μ0.
This implies that Fn0-F∈ℳp′⊂ℳp, and, since Fn0∈ℳp, it follows that F∈ℳp.Proposition 2.8.
ℳ∞=⋃p>0ℳp is a closed  subspace of  ℳ.Proof.
SupposeF0=(f0,i)i∉ℳ∞, then F0+ℳq,q>0 is a neighborhood of F0 and (F0+ℳq)∩ℳp=∅ for all p>0.
Indeed, ifF0+F1=F2 for some F1∈ℳq and some F2∈ℳp, then F0∈ℳr, where r=max(p,q), which is a contradiction. Hence, (F0+ℳq)∩ℳ∞=∅, which implies that ℳ∞ is closed.Remark 2.9.
IfS((0)n,r) denotes the open sphere with center (0)n and radius r, it is easy to see that the family(2.26){S((0)n,r)}r>0∪{F+S((0)n,r)}F∉M∞,r>0
is an open covering  of ℳ without a finite subcovering. Hence ℳ is not compact.



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*Source: 582576-2011-06-01.xml*