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

Abstract

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 ℳ 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 “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 𝑓_𝑛,𝑓∶Γ→ℝ,𝑛=1,2,…, be measurable functions.

We adopt the following usual terminology. By the notation 𝑓→_𝑛𝜇𝑓 we denote that the sequence of measurable functions (𝑓_𝑛)_𝑛 converges in measure to 𝑓. Also for a pair ((𝑓_𝑛)_𝑛,𝑓) and 𝜀≥0 we set 𝐴^𝜀𝑛=𝛾∈Γ∶||𝑓_𝑛(𝛾)−𝑓(𝛾)||≥𝜀=||𝑓_𝑛−𝑓||≥𝜀,𝑛=1,2,….(1.1) We denote by ℕ the set of all positive integers and 𝑐⁺⁰ the set of all real-valued nonnegative sequences (𝜀_𝑛)_𝑛 converging to 0. Also for 𝑝>0 we set ℓ^+𝑝=𝜀_𝑛_𝑛∶𝜀_𝑛≥0for𝑛=1,2,…,^∞_𝑛=1𝜀^𝑝𝑛<∞.(1.2) Convergence in measure is characterized by elements (𝜀_𝑛)_𝑛 of 𝑐⁺⁰ as follows: 𝑓⟶_𝑛𝜇𝑓iffthereexists𝜀_𝑛_𝑛∈𝑐⁺⁰suchthatlim_𝑛→∞𝜇𝐴^𝜀𝑛_𝑛=0.(1.3) Taking into account that the sequence (𝜀_𝑛)_𝑛 above expresses the quality of approximation of (𝑓_𝑛)_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, (𝑓_𝑛)_𝑛𝑝-converges in measure to 𝑓 (and we write 𝑓−−−→_{𝑛𝑝−𝜇}𝑓) if and only if there exists an element (𝜀_𝑛)_𝑛∈ℓ^+𝑝 such that lim_𝑛→∞𝜇𝐴^𝜀𝑛_𝑛=0.(1.4) 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 𝑀₀≨𝑀_𝑝≨𝑀_𝑞≨𝑀_∞≨𝑀,0<𝑝<𝑞,(1.5) where𝑀𝑀=𝑓_𝑛_𝑛∶𝑓⟶_𝑛𝜇0_𝑝,𝑀=𝑓_𝑛_𝑛∶𝑓−−−→_{𝑛𝑝−𝜇}0,𝑝>0,₀=_𝑝>0𝑀_𝑝,𝑀_∞=_𝑝>0𝑀_𝑝.(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={(𝑓_𝑛)_𝑛∶𝑓→_𝑛𝜇0}. 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 (𝑓_𝑛)_𝑛,(𝑔_𝑛)_𝑛 be elements of 𝑀. We say that (𝑓_𝑛)_𝑛, (𝑔_𝑛)_𝑛 are equivalent ((𝑓_𝑛)_𝑛∼(𝑔_𝑛)_𝑛) if and only if for each positive real number 𝑝 there exists an element (𝜀_𝑛)_𝑛 of ℓ^+𝑝 such that lim_𝑛→∞𝜇||𝑓_𝑛−𝑔_𝑛||>𝜀_𝑛=0(2.1) or equivalently 𝑓_𝑛−𝑔−−−→_{𝑛𝑝−𝜇}0,∀𝑝>0⟺𝑓_𝑛−𝑔_𝑛_𝑛∈𝑀₀.(2.2)
Since 𝑀₀ 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 metric 𝑑 on ℳ under which ℳ turns to be a complete metric space, similarly as a Fréchet space.

Definition 2.2. Let (𝑓_𝑛)_𝑛∈ℳ. We define ‖‖(𝑓_𝑛)_𝑛‖‖=arctaninf𝐴𝑓_𝑛,(2.3) where 𝐴𝑓_𝑛=𝑝>0∶𝑓−−−→_{𝑛𝑝−𝜇}0=𝑝>0∶𝑓_𝑛_𝑛∈𝑀_𝑝.(2.4)
By (1.5) it follows that 𝐴(𝑓_𝑛) is an interval in ℝ.

Remarks 2.3. (i) We note that the above set 𝐴(𝑓_𝑛) could be empty. In this case we set ‖(𝑓_𝑛)_𝑛‖=𝜋/2.
(ii) If (𝑓_𝑛)_𝑛∼(𝑔_𝑛)_𝑛, then ‖(𝑓_𝑛)_𝑛‖=‖(𝑔_𝑛)_𝑛‖. (Indeed, as (𝑔_𝑛)_𝑛=(𝑔_𝑛−𝑓_𝑛)_𝑛+(𝑓_𝑛)_𝑛 and (𝑓_𝑛)_𝑛∈𝑀_𝑝, it follows that (𝑓_𝑛)_𝑛∈𝑀_𝑝 if and only if (𝑔_𝑛)_𝑛∈𝑀_𝑝.)
(iii) We set ℳ_𝑝=𝑀_𝑝/∼,𝑝>0, and hence ℳ_𝑝 is a proper vector subspace of ℳ and the following strict inclusion holds: ℳ_𝑝1≨ℳ_𝑝2if0<𝑝₁<𝑝₂(see(1.5)).(2.5)

Proposition 2.4. The function ‖‖∶ℳ→ℝ satisfies the following properties: (i)‖(𝑓_𝑛)_𝑛‖≥0(ii)‖(𝑓_𝑛)_𝑛‖=0 iff (𝑓_𝑛)_𝑛∼(0)_𝑛(iii)‖𝑎(𝑓_𝑛)_𝑛‖=‖(𝑓_𝑛)_𝑛‖for𝑎≠0(iv)‖(𝑓_𝑛+𝑔_𝑛)_𝑛‖≤‖(𝑓_𝑛)_𝑛‖+‖(𝑔_𝑛)_𝑛‖.

Hence, ℳ becomes a metric space and the metric 𝑑((𝑓_𝑛)_𝑛,(𝑔_𝑛)_𝑛)=‖(𝑓_𝑛−𝑔_𝑛)_𝑛‖ is invariant under translations.

Proof. (i) It is obvious.
(ii) If (𝑓_𝑛)_𝑛∼(0)_𝑛, then ‖(𝑓_𝑛)_𝑛‖=0.
Conversely, if ‖(𝑓_𝑛)_𝑛‖=0, then, (𝑓_𝑛)−−−→_{𝑛𝑝−𝜇}0 for each 𝑝>0, and hence (𝑓_𝑛)_𝑛∈ℳ₀ and consequently (𝑓_𝑛)∼(0)_𝑛.
(iii) For 𝑎≠0, it holds that 𝐴^𝜀𝑛_𝑛=||𝑓_𝑛||≥𝜀_𝑛=||𝑎𝑓_𝑛||≥|𝑎|𝜀_𝑛,_∞_𝑛=1𝜀^𝑝𝑛<∞⟺^∞_𝑛=1|𝑎|𝜀_𝑛^𝑝<∞,(2.6) for each sequence (𝜀_𝑛)_𝑛 of positive real numbers. Hence, 𝑓_𝑛−−−→_{𝑛𝑝−𝜇}0iff𝑎𝑓_𝑛−−−→_{𝑛𝑝−𝜇}0,(2.7) which means that ‖(𝑓_𝑛)_𝑛‖=‖(𝑎𝑓_𝑛)_𝑛‖.
(iv) The inequality is obvious if ‖(𝑓_𝑛)_𝑛‖=𝜋/2 or ‖(𝑔_𝑛)_𝑛‖=𝜋/2.
Suppose ‖(𝑓_𝑛)_𝑛‖≤‖(𝑔_𝑛)_𝑛‖<𝜋/2. Then we conclude that 𝐴(𝑔_𝑛)⊂𝐴(𝑓_𝑛). Hence, (𝑔_𝑛)−−−→_{𝑛𝑝−𝜇}0 implies (𝑓_𝑛)_𝑛+(𝑔_𝑛)−−−→_{𝑛𝑝−𝜇}0. So 𝐴(𝑔_𝑛)⊆𝐴(𝑓_𝑛+𝑔_𝑛), which implies that ‖‖𝑓_𝑛+𝑔_𝑛_𝑛‖‖≤‖‖𝑔_𝑛_𝑛‖‖≤‖‖𝑓_𝑛_𝑛‖‖+‖‖𝑔_𝑛_𝑛‖‖.(2.8)

Theorem 2.5. The space (ℳ,𝑑) is a complete metric space.

Proof. Let (𝐹_𝑛)_𝑛 be a Cauchy sequence in ℳ, where 𝐹_𝑛=(𝑓_𝑛,𝑖)_𝑖,𝑛=1,2,…. Hence, there exists an increasing sequence of positive integers (𝑛_𝑘)_𝑘 such that ‖‖𝐹_𝑛−𝐹_𝑚‖‖<arctan1𝑘,for𝑛,𝑚≥𝑛_𝑘,𝑘=1,2,….(2.9) This means that, for each 𝑛,𝑚≥𝑛_𝑘, there exists a sequence (𝜀_{𝑛,𝑚,𝑖})_𝑖 of positive real numbers with ∑^∞𝑖=1𝜀^{1/𝑘𝑛,𝑚,𝑖}<∞ such that 𝜇𝐴_{𝑛,𝑚,𝑖}⟶0,𝑖⟶∞,where𝐴_{𝑛,𝑚,𝑖}=||𝑓_𝑛,𝑖−𝑓_𝑚,𝑖||≥𝜀_{𝑛,𝑚,𝑖}.(2.10) Then, ^∞_𝑖=1max_{𝑛ℓ≤𝑛≤𝑛𝑘+1}𝜀_{𝑛,𝑛𝑘+1,𝑖}^1/ℓ<∞,forℓ=1,2,…,𝑘,𝑘∈ℕ.(2.11) From (2.10) and (2.11), proceeding by induction, it follows that there exists an increasing sequence (𝑖_𝑘)_𝑘 of positive integers such that for each 𝑘 we have 𝜇𝐴_{𝑛,𝑛𝑘+1,𝑖}<1𝑘for𝑖≥𝑖_𝑘,𝑛₁≤𝑛≤𝑛_𝑘+1,𝑘=1,2,…,(2.12)_∞_{𝑖=𝑖𝑘}max_{𝑛ℓ≤𝑛≤𝑛𝑘+1}𝜀_{𝑛,𝑚,𝑖}^1/ℓ<12^𝑘,forℓ=1,2,…,𝑘,(2.13)𝜇||𝑓_{𝑛𝑘+1,𝑖}||≥1𝑘<𝑘1,for𝑖≥𝑖_𝑘,(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 𝐹==𝑓𝑓_𝑛1,1,𝑓_𝑛1,2,…,𝑓_{𝑛1,𝑖1−1};𝑓_𝑛2,𝑖1,…,𝑓_{𝑛2,𝑖2−1};…;𝑓_{𝑛𝑘+1,𝑖𝑘},…,𝑓_{𝑛𝑘+1,𝑖𝑘+1−1};…_𝑖_𝑖.(2.15) By (2.14) it follows that 𝐹=(𝑓_𝑖)_𝑖∈ℳ.
We have to show that ‖‖𝐹_𝑛−𝐹‖‖⟶0,𝑛⟶∞⟺∀ℓ∈ℕ∃𝑛₀∈ℕ∶𝑓_𝑛,𝑖−𝑓_𝑖⟶_{𝑖(1/ℓ)−𝜇}0,for𝑛≥𝑛₀.(2.16) This means that we have to find 𝑛₀∈ℕ and for 𝑛≥𝑛₀ a sequence of positive real numbers (𝜀_𝑖)_𝑖 with ∑^∞𝑖=1𝜀^𝑖1/ℓ<∞ such that 𝜇𝐴_𝑖⟶0,𝑖⟶∞,where𝐴_𝑖=||𝑓_𝑛,𝑖−𝑓_𝑖||≥𝜀_𝑖.(2.17) Indeed let ℓ∈ℕ,𝑛≥𝑛₀=𝑛_ℓ, and 𝑛_ℓ≤𝑛_𝑘<𝑛<𝑛_𝑘+1 for some 𝑘∈ℕ.
We set 𝜀𝜀_𝑖=1,if𝑖=1,2,…,𝑖_𝑘−1,_𝑖𝜀=𝜀_{𝑛,𝑛𝑘+1,𝑖},if𝑖=𝑖_𝑘,…,𝑖_𝑘+1−1,_𝑖=𝜀_{𝑛,𝑛𝑘+2,𝑖},if𝑖=𝑖_𝑘+1,…,𝑖_𝑘+2−1,(2.18) and so on.
It holds that ^∞_𝑖=1𝜀^𝑖1/ℓ≤𝑖_𝑘−1+12^𝑘+12^𝑘+1+⋯<∞(by(2.13)).(2.19) Also, for 𝑖_𝑚≤𝑖≤𝑖_𝑚+1,𝑚≥𝑘, we have that 𝐴_𝑖=||𝑓_𝑛,𝑖−𝑓_𝑖||≥𝜀_𝑖=||𝑓_𝑛,𝑖−𝑓_{𝑛𝑚+1,𝑖}||≥𝜀_{𝑛,𝑛𝑚+1,𝑖}(2.20) (by definition of 𝑓_𝑖). Hence, by (2.12), we take 𝜇𝐴_𝑖<1𝑚.(2.21) This implies (2.17), and the proof is complete.

Remarks 2.6. It is easy to see the following (a)The addition +∶((𝑓_𝑛)_𝑛,(𝑔_𝑛)_𝑛)↦(𝑓_𝑛+𝑔_𝑛)_𝑛 is continuous.(b)The translation 𝑇_{(𝑔𝑛)𝑛}∶(𝑓_𝑛)_𝑛↦(𝑓_𝑛)_𝑛+(𝑔_𝑛)_𝑛=(𝑓_𝑛+𝑔_𝑛)_𝑛 is a homeomorphism. Hence, the system of neighborhoods of (0)_𝑛 determines the topology of (ℳ,𝑑).(c)The multiplication operator 𝐻_𝑎∶𝑓_𝑛_𝑛⟼𝑎⋅𝑓_𝑛_𝑛,𝑎≠0,(2.22) is a homeomorphism.(d)The multiplication (𝑎,(𝑓_𝑛)_𝑛)↦𝑎(𝑓_𝑛)_𝑛 is not continuous. (If 𝑎_𝑛→0,𝑎_𝑛≠0,𝑛=1,2,…, and 𝐹=(𝑓_𝑛)_𝑛∈ℳ, 𝐹≠0, and 𝐹_𝑛=𝐹 for 𝑛=1,2,…, it holds that 𝑎_𝑛⟶𝑎≠0,𝐹⟶_𝑛𝑑𝐹,(2.23) but ‖𝑎_𝑛𝐹_𝑛−0𝐹‖=‖𝑎_𝑛𝐹_𝑛‖=‖𝐹_𝑛‖=‖𝐹‖↛0).(e)The family (ℳ_𝑝)_𝑝>0 is a system of neighborhoods of (0)_𝑛. (Indeed, if 𝑆_𝑟=𝑆((0)_𝑛,𝑟)={(𝑓_𝑛)_𝑛∶‖(𝑓_𝑛)_𝑛‖<𝑟} for 𝑟>0, then for 0<𝑟₁<𝑝<𝑟₂ we have 𝑆_𝑟1⊂ℳ_𝑝⊂𝑆_𝑟2.)
Though (ℳ,𝑑) is not a topological vector space, (ℳ,𝑑) is complete and the subspaces ℳ_𝑝, 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 𝐻_𝑎 is needed (see [11]).

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

Proof. Suppose that 𝑝>0 and (𝐹_𝑛)_𝑛 is a sequence in ℳ_𝑝, where 𝐹_𝑛=(𝑓_𝑛,𝑖)_𝑖, 𝑛=1,2,…, and 𝐹=(𝑓_𝑛)_𝑛∈ℳ such that 𝐹⟶_𝑛𝑑𝐹⟺‖‖𝐹_𝑛−𝐹‖‖⟶0,𝑛→∞.(2.24) Hence, there exist 𝑝′<𝑝 and 𝑛₀ such that 𝐹_𝑛0−𝐹−−−→^{𝑝−𝜇}0.(2.25) This implies that 𝐹_𝑛0−𝐹∈ℳ_𝑝⊂ℳ_𝑝, and, since 𝐹_𝑛0∈ℳ_𝑝, it follows that 𝐹∈ℳ_𝑝.

Proposition 2.8. ℳ_∞=⋃_𝑝>0ℳ_𝑝 is a closed subspace of ℳ.

Proof. Suppose 𝐹₀=(𝑓_0,𝑖)_𝑖∉ℳ_∞, then 𝐹₀+ℳ_𝑞,𝑞>0 is a neighborhood of 𝐹₀ and (𝐹₀+ℳ_𝑞)∩ℳ_𝑝=∅ for all 𝑝>0.
Indeed, if 𝐹₀+𝐹₁=𝐹₂ for some 𝐹₁∈ℳ_𝑞 and some 𝐹₂∈ℳ_𝑝, then 𝐹₀∈ℳ_𝑟, where 𝑟=max(𝑝,𝑞), which is a contradiction. Hence, (𝐹₀+ℳ_𝑞)∩ℳ_∞=∅, which implies that ℳ_∞ is closed.

Remark 2.9. If S((0)_𝑛,𝑟) denotes the open sphere with center (0)_𝑛 and radius 𝑟, it is easy to see that the family𝑆(0)_𝑛,𝑟_𝑟>0∪𝐹+𝑆(0)_𝑛,𝑟_{𝐹∉ℳ∞},𝑟>0(2.26) is an open covering of ℳ without a finite subcovering. Hence ℳ is not compact.

Acknowledgment

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