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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 582576, 7 pages
Research Article

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

1Department of Mathematics and Computer Sciences, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
2Department 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.


In the study by Papanastassiou and Papachristodoulos, 2009 the notion of 𝑝-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 β„³ 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 β€œBochner-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 HaluΕ‘ka and HutnΓ­k [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 (Ξ“,Ξ£,πœ‡) be an arbitrary measure space, where πœ‡ is a [0,∞]-valued measure, and let 𝑓𝑛,π‘“βˆΆΞ“β†’β„,𝑛=1,2,…, be measurable functions.

We adopt the following usual terminology. By the notation π‘“π‘›πœ‡β†’π‘“ we denote that the sequence of measurable functions (𝑓𝑛)𝑛 converges in measure to 𝑓. Also for a pair ((𝑓𝑛)𝑛,𝑓) and πœ€β‰₯0 we set π΄πœ€π‘›=ξ€½||π‘“π›ΎβˆˆΞ“βˆΆπ‘›||ξ€Ύ=ξ€½||𝑓(𝛾)βˆ’π‘“(𝛾)β‰₯πœ€π‘›||ξ€Ύβˆ’π‘“β‰₯πœ€,𝑛=1,2,….(1.1) We denote by β„• the set of all positive integers and 𝑐+0 the set of all real-valued nonnegative sequences (πœ€π‘›)𝑛 converging to 0. Also for 𝑝>0 we set β„“+𝑝=ξƒ―ξ€·πœ€π‘›ξ€Έπ‘›βˆΆπœ€π‘›β‰₯0for𝑛=1,2,…,βˆžξ“π‘›=1πœ€π‘π‘›ξƒ°.<∞(1.2) Convergence in measure is characterized by elements (πœ€π‘›)𝑛 of 𝑐+0 as follows: π‘“π‘›πœ‡βŸΆπ‘“iffthereexistsξ€·πœ€π‘›ξ€Έπ‘›βˆˆπ‘+0suchthatlimπ‘›β†’βˆžπœ‡ξ€·π΄πœ€π‘›π‘›ξ€Έ=0.(1.3) Taking into account that the sequence (πœ€π‘›)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, given 𝑝>0, (𝑓𝑛)𝑛𝑝-converges in measure to 𝑓 (and we write π‘“π‘›π‘βˆ’πœ‡βˆ’βˆ’βˆ’β†’π‘“) if and only if there exists an element (πœ€π‘›)π‘›βˆˆβ„“+𝑝 such that limπ‘›β†’βˆžπœ‡ξ€·π΄πœ€π‘›π‘›ξ€Έ=0.(1.4) Obviously 𝑝-convergence in measure implies convergence in measure. It is proved (see [10, Preposition 2.3]) that if the measure πœ‡ is not trivial and 0<𝑝<π‘ž, then 𝑝-convergence in measure implies π‘ž-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 𝑀0β‰¨π‘€π‘β‰¨π‘€π‘žβ‰¨π‘€βˆžβ‰¨π‘€,0<𝑝<π‘ž,(1.5) where𝑓𝑀=π‘›ξ€Έπ‘›βˆΆπ‘“π‘›πœ‡ξ‚‡,π‘€βŸΆ0𝑝=ξ‚†ξ€·π‘“π‘›ξ€Έπ‘›βˆΆπ‘“π‘›π‘βˆ’πœ‡ξ‚‡π‘€βˆ’βˆ’βˆ’β†’0,𝑝>0,0=𝑝>0𝑀𝑝,π‘€βˆž=ξšπ‘>0𝑀𝑝.(1.6) 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 way 𝑝-convergence in measure induces an equivalence relation on the vector space 𝑀={(𝑓𝑛)π‘›βˆΆπ‘“π‘›πœ‡β†’0}. We consider 𝑀 as a subspace of 𝐿0(Ξ“)β„•, β„΅0 copies of the vector space 𝐿0(Ξ“) 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)π‘›βˆΌ(𝑔𝑛)𝑛) if and only if for each positive real number 𝑝 there exists an element (πœ€π‘›)𝑛 of β„“+𝑝 such that limπ‘›β†’βˆžπœ‡||π‘“ξ€·ξ€Ίπ‘›βˆ’π‘”π‘›||>πœ€π‘›ξ€»ξ€Έ=0(2.1) or equivalently π‘“π‘›βˆ’π‘”π‘›π‘βˆ’πœ‡ξ€·π‘“βˆ’βˆ’βˆ’β†’0,βˆ€π‘>0βŸΊπ‘›βˆ’π‘”π‘›ξ€Έπ‘›βˆˆπ‘€0.(2.2)
Since 𝑀0 is a vector subspace of 𝑀 the relation ~ is an equivalence one. We set β„³=π‘€βˆΌ=𝑀/𝑀0.
In the sequel we will define a metric 𝑑 on β„³ under which β„³ turns to be a complete metric space, similarly as a FrΓ©chet space.

Definition 2.2. Let (𝑓𝑛)π‘›βˆˆβ„³. We define β€–β€–(𝑓𝑛)𝑛‖‖𝑓=arctaninf𝐴𝑛,ξ€Έξ€Έ(2.3) where 𝐴𝑓𝑛=𝑝>0βˆΆπ‘“π‘›π‘βˆ’πœ‡ξ‚‡=ξ€½ξ€·π‘“βˆ’βˆ’βˆ’β†’0𝑝>0βˆΆπ‘›ξ€Έπ‘›βˆˆπ‘€π‘ξ€Ύ.(2.4)
By (1.5) it follows that 𝐴(𝑓𝑛) is an interval in ℝ.

Remarks 2.3. (i) We note that the above set 𝐴(𝑓𝑛) could be empty. In this case we set β€–(𝑓𝑛)𝑛‖=πœ‹/2.
(ii) If (𝑓𝑛)π‘›βˆΌ(𝑔𝑛)𝑛, then β€–(𝑓𝑛)𝑛‖=β€–(𝑔𝑛)𝑛‖. (Indeed, as (𝑔𝑛)𝑛=(π‘”π‘›βˆ’π‘“π‘›)𝑛+(𝑓𝑛)𝑛 and (𝑓𝑛)π‘›βˆˆπ‘€π‘, it follows that (𝑓𝑛)π‘›βˆˆπ‘€π‘ if and only if (𝑔𝑛)π‘›βˆˆπ‘€π‘.)
(iii) We set ℳ𝑝=𝑀𝑝/∼,𝑝>0, and hence ℳ𝑝 is a proper vector subspace of β„³ and the following strict inclusion holds: ℳ𝑝1≨ℳ𝑝2if0<𝑝1<𝑝2(see(1.5)).(2.5)

Proposition 2.4. The function β€–β€–βˆΆβ„³β†’β„ satisfies the following properties: (i)β€–(𝑓𝑛)𝑛‖β‰₯0(ii)β€–(𝑓𝑛)𝑛‖=0 iff (𝑓𝑛)π‘›βˆΌ(0)𝑛(iii)β€–π‘Ž(𝑓𝑛)𝑛‖=β€–(𝑓𝑛)𝑛‖forπ‘Žβ‰ 0(iv)β€–(𝑓𝑛+𝑔𝑛)𝑛‖≀‖(𝑓𝑛)𝑛‖+β€–(𝑔𝑛)𝑛‖.

Hence, β„³ becomes a metric space and the metric 𝑑((𝑓𝑛)𝑛,(𝑔𝑛)𝑛)=β€–(π‘“π‘›βˆ’π‘”π‘›)𝑛‖ is invariant under translations.

Proof. (i) It is obvious.
(ii) If (𝑓𝑛)π‘›βˆΌ(0)𝑛, then β€–(𝑓𝑛)𝑛‖=0.
Conversely, if β€–(𝑓𝑛)𝑛‖=0, then, (𝑓𝑛)π‘›π‘βˆ’πœ‡βˆ’βˆ’βˆ’β†’0 for each 𝑝>0, and hence (𝑓𝑛)π‘›βˆˆβ„³0 and consequently (𝑓𝑛)∼(0)𝑛.
(iii) For π‘Žβ‰ 0, it holds that π΄πœ€π‘›π‘›=ξ€Ί||𝑓𝑛||β‰₯πœ€π‘›ξ€»=ξ€Ί||π‘Žπ‘“π‘›||β‰₯|π‘Ž|πœ€π‘›ξ€»,βˆžξ“π‘›=1πœ€π‘π‘›<βˆžβŸΊβˆžξ“π‘›=1ξ€·|π‘Ž|πœ€π‘›ξ€Έπ‘<∞,(2.6) for each sequence (πœ€π‘›)𝑛 of positive real numbers. Hence, ξ€·π‘“π‘›ξ€Έπ‘›π‘βˆ’πœ‡βˆ’βˆ’βˆ’β†’0iο¬€ξ€·π‘Žπ‘“π‘›ξ€Έπ‘›π‘βˆ’πœ‡βˆ’βˆ’βˆ’β†’0,(2.7) which means that β€–(𝑓𝑛)𝑛‖=β€–(π‘Žπ‘“π‘›)𝑛‖.
(iv) The inequality is obvious if β€–(𝑓𝑛)𝑛‖=πœ‹/2 or β€–(𝑔𝑛)𝑛‖=πœ‹/2.
Suppose β€–(𝑓𝑛)𝑛‖≀‖(𝑔𝑛)𝑛‖<πœ‹/2. Then we conclude that 𝐴(𝑔𝑛)βŠ‚π΄(𝑓𝑛). Hence, (𝑔𝑛)π‘›π‘βˆ’πœ‡βˆ’βˆ’βˆ’β†’0 implies (𝑓𝑛)𝑛+(𝑔𝑛)π‘›π‘βˆ’πœ‡βˆ’βˆ’βˆ’β†’0. So 𝐴(𝑔𝑛)βŠ†π΄(𝑓𝑛+𝑔𝑛), which implies that ‖‖𝑓𝑛+𝑔𝑛𝑛‖‖≀‖‖𝑔𝑛𝑛‖‖≀‖‖𝑓𝑛𝑛‖‖+‖‖𝑔𝑛𝑛‖‖.(2.8)

Theorem 2.5. The space (β„³,𝑑) is a complete metric space.

Proof. Let (𝐹𝑛)𝑛 be a Cauchy sequence in β„³, where 𝐹𝑛=(𝑓𝑛,𝑖)𝑖,𝑛=1,2,…. Hence, there exists an increasing sequence of positive integers (π‘›π‘˜)π‘˜ such that β€–β€–πΉπ‘›βˆ’πΉπ‘šβ€–β€–1<arctanπ‘˜,for𝑛,π‘šβ‰₯π‘›π‘˜,π‘˜=1,2,….(2.9) This means that, for each 𝑛,π‘šβ‰₯π‘›π‘˜, there exists a sequence (πœ€π‘›,π‘š,𝑖)𝑖 of positive real numbers with βˆ‘βˆžπ‘–=1πœ€1/π‘˜π‘›,π‘š,𝑖<∞ such that πœ‡ξ€·π΄π‘›,π‘š,π‘–ξ€ΈβŸΆ0,π‘–βŸΆβˆž,where𝐴𝑛,π‘š,𝑖=ξ€Ί||𝑓𝑛,π‘–βˆ’π‘“π‘š,𝑖||β‰₯πœ€π‘›,π‘š,𝑖.(2.10) Then, βˆžξ“π‘–=1ξ‚΅maxπ‘›β„“β‰€π‘›β‰€π‘›π‘˜+1πœ€π‘›,π‘›π‘˜+1,𝑖1/β„“<∞,forβ„“=1,2,…,π‘˜,π‘˜βˆˆβ„•.(2.11) From (2.10) and (2.11), proceeding by induction, it follows that there exists an increasing sequence (π‘–π‘˜)π‘˜ of positive integers such that for each π‘˜ we have πœ‡ξ‚€π΄π‘›,π‘›π‘˜+1,𝑖<1π‘˜for𝑖β‰₯π‘–π‘˜,𝑛1β‰€π‘›β‰€π‘›π‘˜+1,π‘˜=1,2,…,(2.12)βˆžξ“π‘–=π‘–π‘˜ξ‚΅maxπ‘›β„“β‰€π‘›β‰€π‘›π‘˜+1πœ€π‘›,π‘š,𝑖1/β„“<12π‘˜,forβ„“=1,2,…,π‘˜,(2.13)πœ‡ξ‚ƒ||π‘“π‘›π‘˜+1,𝑖||β‰₯1π‘˜ξ‚„<1π‘˜,for𝑖β‰₯π‘–π‘˜,(2.14)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 𝑓𝐹=𝑛1,1,𝑓𝑛1,2,…,𝑓𝑛1,𝑖1βˆ’1;𝑓𝑛2,𝑖1,…,𝑓𝑛2,𝑖2βˆ’1;…;π‘“π‘›π‘˜+1,π‘–π‘˜,…,π‘“π‘›π‘˜+1,π‘–π‘˜+1βˆ’1=𝑓;…𝑖𝑖.(2.15) By (2.14) it follows that 𝐹=(𝑓𝑖)π‘–βˆˆβ„³.
We have to show that β€–β€–πΉπ‘›β€–β€–βˆ’πΉβŸΆ0,π‘›βŸΆβˆžβŸΊβˆ€β„“βˆˆβ„•βˆƒπ‘›0ξ€·π‘“βˆˆβ„•βˆΆπ‘›,π‘–βˆ’π‘“π‘–ξ€Έπ‘–(1/β„“)βˆ’πœ‡βŸΆ0,for𝑛β‰₯𝑛0.(2.16) This means that we have to find 𝑛0βˆˆβ„• and for 𝑛β‰₯𝑛0 a sequence of positive real numbers (πœ€π‘–)𝑖 with βˆ‘βˆžπ‘–=1πœ€π‘–1/β„“<∞ such that πœ‡ξ€·π΄π‘–ξ€ΈβŸΆ0,π‘–βŸΆβˆž,where𝐴𝑖=ξ€Ί||𝑓𝑛,π‘–βˆ’π‘“π‘–||β‰₯πœ€π‘–ξ€».(2.17) Indeed let β„“βˆˆβ„•,𝑛β‰₯𝑛0=𝑛ℓ, and π‘›β„“β‰€π‘›π‘˜<𝑛<π‘›π‘˜+1 for some π‘˜βˆˆβ„•.
We set πœ€π‘–=1,if𝑖=1,2,…,π‘–π‘˜πœ€βˆ’1,𝑖=πœ€π‘›,π‘›π‘˜+1,𝑖,if𝑖=π‘–π‘˜,…,π‘–π‘˜+1πœ€βˆ’1,𝑖=πœ€π‘›,π‘›π‘˜+2,𝑖,if𝑖=π‘–π‘˜+1,…,π‘–π‘˜+2βˆ’1,(2.18) and so on.
It holds that βˆžξ“π‘–=1πœ€π‘–1/β„“β‰€ξ€·π‘–π‘˜ξ€Έ+1βˆ’12π‘˜+12π‘˜+1+β‹―<∞(by(2.13)).(2.19) Also, for π‘–π‘šβ‰€π‘–β‰€π‘–π‘š+1,π‘šβ‰₯π‘˜, we have that 𝐴𝑖=ξ€Ί||𝑓𝑛,π‘–βˆ’π‘“π‘–||β‰₯πœ€π‘–ξ€»=||𝑓𝑛,π‘–βˆ’π‘“π‘›π‘š+1,𝑖||β‰₯πœ€π‘›,π‘›π‘š+1,𝑖(2.20) (by definition of 𝑓𝑖). Hence, by (2.12), we take πœ‡ξ€·π΄π‘–ξ€Έ<1π‘š.(2.21) 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 translation 𝑇(𝑔𝑛)π‘›βˆΆ(𝑓𝑛)𝑛↦(𝑓𝑛)𝑛+(𝑔𝑛)𝑛=(𝑓𝑛+𝑔𝑛)𝑛 is a homeomorphism. Hence, the system of neighborhoods of (0)𝑛 determines the topology of (β„³,𝑑).(c)The multiplication operator π»π‘ŽβˆΆξ€·π‘“π‘›ξ€Έπ‘›ξ€·π‘“βŸΌπ‘Žβ‹…π‘›ξ€Έπ‘›,π‘Žβ‰ 0,(2.22) is a homeomorphism.(d)The multiplication (π‘Ž,(𝑓𝑛)𝑛)β†¦π‘Ž(𝑓𝑛)𝑛 is not continuous. (If π‘Žπ‘›β†’0,π‘Žπ‘›β‰ 0,𝑛=1,2,…, and 𝐹=(𝑓𝑛)π‘›βˆˆβ„³, 𝐹≠0, and 𝐹𝑛=𝐹 for 𝑛=1,2,…, it holds that π‘Žπ‘›βŸΆπ‘Žβ‰ 0,πΉπ‘›π‘‘βŸΆπΉ,(2.23) but β€–π‘Žπ‘›πΉπ‘›βˆ’0𝐹‖=β€–π‘Žπ‘›πΉπ‘›β€–=‖𝐹𝑛‖=‖𝐹‖↛0).(e)The family (ℳ𝑝)𝑝>0 is a system of neighborhoods of (0)𝑛. (Indeed, if π‘†π‘Ÿ=𝑆((0)𝑛,π‘Ÿ)={(𝑓𝑛)π‘›βˆΆβ€–(𝑓𝑛)𝑛‖<π‘Ÿ} for π‘Ÿ>0, then for 0<π‘Ÿ1<𝑝<π‘Ÿ2 we have π‘†π‘Ÿ1βŠ‚β„³π‘βŠ‚π‘†π‘Ÿ2.)
Though (β„³,𝑑) is not a topological vector space, (β„³,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, (β„³,𝑑) is something like a FrΓ©chet space. For example, the principle of uniform boundedness holds true, as for this principle only continuity of π»π‘Ž is needed (see [11]).

Proposition 2.7. The subspaces ℳ𝑝 are closed for each 𝑝>0.

Proof. Suppose that 𝑝>0 and (𝐹𝑛)𝑛 is a sequence in ℳ𝑝, where 𝐹𝑛=(𝑓𝑛,𝑖)𝑖, 𝑛=1,2,…, and 𝐹=(𝑓𝑛)π‘›βˆˆβ„³ such that πΉπ‘›π‘‘β€–β€–πΉβŸΆπΉβŸΊπ‘›β€–β€–βˆ’πΉβŸΆ0,π‘›β†’βˆž.(2.24) Hence, there exist 𝑝′<𝑝 and 𝑛0 such that 𝐹𝑛0βˆ’πΉπ‘ξ…žβˆ’πœ‡βˆ’βˆ’βˆ’β†’0.(2.25) This implies that 𝐹𝑛0βˆ’πΉβˆˆβ„³π‘ξ…žβŠ‚β„³π‘, and, since 𝐹𝑛0βˆˆβ„³π‘, it follows that πΉβˆˆβ„³π‘.

Proposition 2.8. β„³βˆž=⋃𝑝>0ℳ𝑝 is a closed subspace of β„³.

Proof. Suppose 𝐹0=(𝑓0,𝑖)π‘–βˆ‰β„³βˆž, then 𝐹0+β„³π‘ž,π‘ž>0 is a neighborhood of 𝐹0 and (𝐹0+β„³π‘ž)βˆ©β„³π‘=βˆ… for all 𝑝>0.
Indeed, if 𝐹0+𝐹1=𝐹2 for some 𝐹1βˆˆβ„³π‘ž and some 𝐹2βˆˆβ„³π‘, then 𝐹0βˆˆβ„³π‘Ÿ, where π‘Ÿ=max(𝑝,π‘ž), which is a contradiction. Hence, (𝐹0+β„³π‘ž)βˆ©β„³βˆž=βˆ…, which implies that β„³βˆž is closed.

Remark 2.9. If S((0)𝑛,π‘Ÿ) denotes the open sphere with center (0)𝑛 and radius π‘Ÿ, it is easy to see that the family𝑆(0)𝑛,π‘Ÿξ€Έξ€Ύπ‘Ÿ>0βˆͺ𝐹+𝑆(0)𝑛,π‘Ÿξ€Έξ€ΎπΉβˆ‰β„³βˆž,π‘Ÿ>0(2.26) is an open covering of β„³ without a finite subcovering. Hence β„³ is not compact.


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


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