# Egoroff’s Theorem and Lusin’s Theorem for Capacities in the Framework ofg-Expectation

Authors: Zhaojun Zong; Feng Hu; Xiaoxin Tian
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In the classical real analysis theory, Egoroff’s theorem and Lusin’s theorem are two of the most important theorems. Theσ-additivity of measures plays a crucial role in the proofs of these theorems. Later, many researchers have carried out lots of studies on Egoroff’s theorem and Lusin’s theorem when the measure is monotone and nonadditive (see, e.g., Li and Yasuda (2004) and Li and Mesiar (2011)). In this paper, we study Egoroff’s theorem and Lusin’s theorem for capacities in the framework of g-expectation. We give some different assumptions that provide Egoroff’s theorem and Lusin’s theorem in the framework of g-expectation.


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## Body

## 1. Introduction

In the classical real analysis theory, Egoroff’s theorem and Lusin’s theorem are two of the most important theorems. Theσ-additivity of measures plays a crucial role in the proofs of these theorems. But in fact, the σ-additivity of measures has been abandoned in some areas because many uncertain phenomena cannot be well modelled by using additive measures.The research studies on Egoroff’s theorem in nonadditive measure theory were carried out by Wang and Klir [1]; Li [2]; Li and Yasuda [3]; and Murofushi et al. [4]. These results faithfully contribute to nonadditive measure theory. Li [2] introduced the concept of condition (E) of set function and proved an essential result: a necessary and sufficient condition that Egoroff’s theorem remains valid for monotone set function is that the monotone set function fulfils condition (E). Murofushi et al. [4] defined the concept of Egoroff condition and proved that it is a necessary and sufficient condition for Egoroff’s theorem with respect to nonadditive measures. Li and Yasuda [3] studied Egoroff’s theorem on finite monotone nonadditive measure space by using condition (E).In nonadditive measure theory, Lusin’s theorem was generalized by Wu and Ha [5] under the conditions of continuity and autocontinuity. Further research on this matter was performed by Jiang and Suzuki [6]. Kawabe [7] investigated regularity and Lusin’s theorem for Riesz space-valued fuzzy measures. Li and Mesiar [8] proved Lusin’s theorem on monotone measure spaces, assuming that the monotone measure fulfils condition (E) and has p.g.p. that was introduced by Dobrakov and Farkova [9].The original motivation for studying nonlinear expectation andg-expectation comes from expected utility theory, which is the foundation of modern mathematical economics. Chen and Epstein [10] gave an application of dynamically consistent nonlinear expectation to recursive utility. Peng [11, 12] and Rosazza Gianin [13] investigated some applications of dynamically consistent nonlinear expectations and g-expectations to static and dynamic pricing mechanisms and risk measures. Hu et al. [14] studied Fubini’s theorem for nonadditive measures in the framework of g-expectation.In this paper, we study Egoroff’s theorem and Lusin’s theorem for capacities induced byg-expectation. We give the sufficient conditions that provide Egoroff’s theorem and Lusin’s theorem in the framework of g-expectation. The remainder of this paper is organized as follows: In Section 2, we introduce some notations, assumptions, notions, lemmas, and propositions that are used in this paper. In Section 3, we give Egoroff’s theorem, Lusin’s theorem, and continuous function approximation theorem in the framework of g-expectation including the proofs.


## 2. Preliminaries

In this section, we shall present some notations, assumptions, notions, lemmas, and propositions that are used in this paper.LetΩ,ℱ,P be a complete probability space and Wtt≥0 be a d-dimensional standard Brownian motion with respect to filtration ℱtt≥0 generated by the Brownian motion and all P-null subsets, i.e.,(1)ℱt=σWs;s≤t∨N,where N is the set of all P-null subsets. Fix a real number T>0.Let us introduce the following spaces:L2Ω,ℱT,P=ξ:ξisFT−measurable random variablesuchthatEξ2<∞L20,T;P;ℝd=V:Vtisℝd‐valued andℱt‐adapted process such thatE∫0TVt2dt<∞S20,T;P;ℝ=V:Vtis continuous process inL20,T;P;ℝsuch thatEsup0≤t≤TVt2<∞Now, we consider the following 1-dimensional backward stochastic differential equation (BSDE):(2)yt=ξ+∫tTgt,ys,zsds−∫tTzsdWs,t∈0,T.Let(3)g:Ω×0,T×ℝ×ℝd⟼ℝ,such that for any y,z∈ℝ×ℝd, g⋅,y,z is ℱt-progressively measurable. We make the following assumptions:(H1)E∫0Tgt,0,02dt<∞.(H2) There exists a constantμ>0 such that for any ω∈Ω,t∈0,T,y1,y2∈ℝ and z1,z2∈ℝd,(4)gt,y1,z1−gt,y2,z2≤μy1−y2+z1−z2.(H3) For anyω∈Ω,t∈0,T and y∈ℝ, gt,y,0=0.(H4)g is subadditive with respect to y and z, i.e., for any ω∈Ω,t∈0,T,y1,y2∈ℝ and z1,z2∈ℝd,(5)gt,y1+y2,z1+z2≤gt,y1,z1+gt,y2,z2.Lemma 1 (see Pardoux and Peng [15]).
Suppose thatg satisfies (H1) and (H2). Then, for any ξ∈L2Ω,ℱT,P, BSDE (2) has a unique pair of adapted processes yt,zt∈S20,T;P;ℝ×L20,T;P;ℝd.Definition 1 (g-expectation, see Peng [16]).
Suppose thatg satisfies (H2) and (H3). For any ξ∈L2Ω,ℱT,P, let yt,zt be the solution of BSDE (2) with terminal value ξ. Consider the mapping εg⋅:L2Ω,ℱT,P⟼ℝ, denoted by εgξ=y0. We call εgξ the g-expectation of ξ.
From Peng [16], we know that that g-expectation keeps many properties of mathematical expectation:(i)
εgc=c, if c is a constant(ii)
εgξ1≥εgξ2, if ξ1≥ξ2
For more details of the properties ofg-expectation, we can see Briand et al. [17]; Chen et al. [18, 19]; Jiang [20]; He et al. [21]; Hu [22]; Zong and Hu [23, 24]; and Zong et al. [25].Proposition 1 (see Briand et al.
[17]). Suppose that g satisfies (H2) and (H3). For any ξ,η∈L2Ω,ℱT,P, there exists a positive constant C such that(6)εgξ−εgη2≤CEξ−η2.Definition 2 (see Choquet [26]).
A capacity is a real-valued set functionV:ℱ⟼0,1 satisfying(1)
V∅=0,VΩ=1(2)
VA≤VB, whenever and A,B∈ℱ
Define the conjugateV¯ of V by V¯A:=1−VΩ\A, ∀A∈ℱ. Obviously, V¯ is also a capacity and V¯¯=V.Definition 3.
Suppose thatV is a capacity. Then,(i)
Countably subadditive:(7)V∪n=1∞An≤∑n=1∞VAn,∀An∈ℱ.(ii)
Continuity from above: for anyAn,A∈ℱ n=1,2,…, limn⟶∞VAn=VA, whenever An↘A.(iii)
Continuity from below: for anyAn,A∈ℱ n=1,2,…, limn⟶∞VAn=VA, whenever An↗A.(iv)
Continuity:V is continuous from below and above.Definition 4 (see Wang and Klir [1]).
LetF be the class of all finite real-valued measurable functions on Ω,ℱ,V, and let f,fn∈F n=1,2,…:(i)
fn converges almost everywhere to f on Ω fn⟶a.e.f: there is a set E∈ℱ such that VE=0 and fn⟶f on Ω\E(ii)
fn converges pseudo almost everywhere to f on Ω (fn⟶p.a.e.f): there is a set Q∈ℱ, such that VΩ\Q=1 and fn⟶f on Ω\Q(iii)
fn converges almost uniformly to f on Ω (fn⟶a.u.f): for any δ>0, there is a set Eδ∈ℱ, such that VΩ\Eδ<δ and fn converges to f uniformly on Eδ(iv)
fn converges to f pseudo almost uniformly on Ω fn⟶p.a.u.f: there exists Qk⊂ℱ with limk⟶∞VΩ\Qk=1 such that fn converges to f on Ω\Qk uniformly for any fixed k=1,2,…Remark 1.
It is easy to prove that(1)
fn⟶a.e.f with respect to V if and only if fn⟶p.a.e.f with respect to V¯(2)
fn⟶a.u.f with respect to V if and only if fn⟶p.a.u.f with respect to V¯
Define(8)VgA≔εgIA,∀A∈ℱT.
It is easy to check thatVg is a capacity.Remark 2.
By Proposition1, we can obtain that suppose g satisfies (H2) and (H3), An,A∈ℱT n=1,2,…; then(1)
VgAn↘VgA, whenever An↘A(2)
VgAn↗VgA, whenever An↗A
Thus,Vg is a continuous capacity. Similarly, Vg¯ is a continuous capacity.
The following proposition is a special case of Corollary 3.5 by Peng [12].Proposition 2.
Suppose thatg satisfies (H2)–(H4). Then, VgA1∪A2≤VgA1+VgA2 ∀A1,A2∈ℱT.Remark 3.
Suppose thatg satisfies (H2)–(H4). By Remark 2 and Proposition 2, we have(9)Vg∪k=1∞Ak=limn⟶∞Vg∪k=1nAk≤limn⟶∞∑k=1nVgAk=∑k=1∞VgAk.
Thus,Vg is countably subadditive.


## 3. Main Results

In this section, we study Egoroff’s theorem, Lusin’s theorem, and continuous function approximation theorem in the framework ofg-expectation.Theorem 1 (Egoroff’s Theorem).
Suppose thatg satisfies (H2)–(H4), fn and f are ℱT-measurable random variables. Then,(1)
Iffn⟶a.e.f with respect to Vg, then fn⟶a.u.f with respect to Vg(2)
Iffn⟶p.a.e.f with respect to V¯g, then fn⟶p.a.u.f with respect to V¯gProof.
Firstly, we prove Theorem1 (1). Let D be the set of these points ω at which fn does not converge to f. Then,(10)D=∪k=1∞∩N=1∞∪n=N∞ω:fnω−fω≥1k.
Sincefn⟶a.e.f with respect to Vg, we have VgD=0. Thus, for any fixed positive integer k,(11)Vg∩N=1∞∪n=N∞ω:fnω−fω≥1k=0.
Noting the fact that(12)∪n=N∞ω:fnω−fω≥1k↘∩N=1∞∪n=N∞ω:fnω−fω≥1k,and by Remark 2, we have(13)limN⟶∞Vg∪n=N∞ω:fnω−fω≥1k=Vg∩N=1∞∪n=N∞ω:fnω−fω≥1k=0.
Therefore for anyδ>0 and any positive integer k, there exists a positive integer Nk, such that(14)Vg∪n=Nk∞ω:fnω−fω≥1k<δ2k.
Let(15)Eδ≔∩k=1∞∩n=Nk∞ω:fnω−fω<1k.
By Remark3, we have(16)VgΩ\Eδ=Vg∪k=1∞∪n=Nk∞ω:fnω−fω≥1k≤∑k=1∞Vg∪n=Nk∞ω:fnω−fω≥1k<∑k=1∞δ2k=δ.
Thus,fn converges to f uniformly on Eδ. The proof of Theorem 1 (1) is complete.
From Theorem1 (1) and by Remark 1, we can easily obtain Theorem 1 (2).
From now on, for studying Lusin’s theorem, we consider the following path spaces:Ω=C0dℝ+ is the space of all ℝd-valued continuous paths ωtt≥0 with ω0=0, equipped with the distance(17)ρω1,ω2≔∑n=1∞2−nmaxt∈0,nωt1−ωt2∧1.
We setΩT:=ω⋅∧T:ω∈Ω. It is clear that Ω,ρ and ΩT,ρ are both complete separable metric spaces. Let O and C be the classes of open sets and closed sets in Ω,ρ, respectively. Similarly, OT and CT are the classes of open sets and closed sets in ΩT,ρ, respectively.
We consider the canonical process:ωt=Wtω,t∈0,∞, for ω∈Ω. Let ℱ be the smallest σ-algebra containing O, and let ℱT be the smallest σ-algebra containing OT. We can choose a probability measure P˜ such that Wtt≥0 is a d-dimensional standard Brownian motion under C0dℝ+,ℱ,P˜.Definition 5 (see Wu and Ha [5]).
A capacityV is called regular, if for every A∈ℱ and δ>0, there exists a closed set Fδ and an open set Gδ of Ω, such that(18)Fδ⊂A⊂Gδ,VGδ\Fδ<δ.Lemma 2.
Suppose thatg satisfies (H2)–(H4), then Vg is regular on ℱT.Proof.
LetA be the class of all sets A∈ℱT such that for any δ>0, there exists a closed set Fδ and an open set Gδ of ΩT satisfying(19)Fδ⊂A⊂Gδ,VgGδ\Fδ<δ.
To prove this lemma, it is sufficient to show thatℱT⊂A.
Firstly, we verify thatA is an algebra. It is easy to know that ΩT∈A. Suppose A,B∈A, then for any δ>0, there exist closed sets F1,δ,F2,δ∈ΩT and open sets G1,δ,G2,δ∈ΩT such that(20)F1,δ⊂A⊂G1,δ,VgG1,δ\F1,δ<δ;F2,δ⊂B⊂G2,δ,VgG2,δ\F2,δ<δ.
So we have(21)F1,δG2,δc⊂A\B⊂G1,δF2,δc.F1,δG2,δc is a closed set of ΩT, G1,δF2,δc is an open set of ΩT, and(22)VgG1,δF2,δc\F1,δG2,δc=VgG1,δF2,δcF1,δcG2,δ=VgG1,δ\F1,δ∩G2,δ\F2,δ≤minVgG1,δ\F1,δ,VgG2,δ\F2,δ<δ.That is, A\B⊂A. So A is an algebra of ΩT.
Next, we prove thatA is closed under the formation of pairwise disjoint countable unions. Let Ann=1∞⊂A be the sequence of pairwise disjoint set and δ>0 be given. From the definition of A and An∈A, we know that for each given n, there exist an open set Gn and a closed set Fn of ΩT such that(23)Fn⊂An⊂Gn,VgGn\Fn<δ2n+1.
Noting the fact that(24)∪n=1∞Fn\∪n=1kFn↘∅,and by Remark 2, we have(25)limk⟶∞Vg∪n=1∞Fn\∪n=1kFn=0.
Thus, there exists a positive integerk0 such that(26)Vg∪n=1∞Fn\∪n=1kFn<δ2.
DenoteGδ:=∪n=1∞Gn and Fδ:=∪n=1k0Fn; then, Gδ is an open set of ΩT, Fδ is a closed set of ΩT, and(27)Fδ⊂∪n=1∞An⊂Gδ.
By Remark3, we have(28)VgGδ\Fδ=Vg∪n=1∞Fn\∪n=1k0Fn≤Vg∪n=1∞Gn\Fn∪∪n=1∞Fn\∪n=1k0Fn≤Vg∪n=1∞Gn\Fn+Vg∪n=1∞Fn\∪n=1k0Fn≤∑n=1∞VgGn\Fn+Vg∪n=1∞Fn\∪n=1k0Fn<δ.
That is,(29)∪n=1∞An∈A.
SoA is a σ-algebra of ΩT.
In real analysis theory, we know that for any closed setF∈CT, there exists a sequence of open sets Enn=1∞ such that(30)En\F↘∅,asn⟶∞.
Therefore, by Remark2, we have limn⟶∞VgEn\F=0. Thus, CT⊂A. Since A is closed under the formation of complements, we have OT⊂A. This shows that A is a σ-algebra containing OT. So ℱT⊂A.Remark 4.
Suppose thatg satisfies (H2)–(H4).(1)
By Lemma2, we know that for any A∈ℱT, there exist an increasing sequence Fnn=1∞ of closed sets and a decreasing sequence Gnn=1∞ of open sets such that for every n=1,2,…, Fn⊂A⊂Gm,(31)VgGn\A<1n,VgA\Fn<1n.(2)
By Theorem1 (1) and Lemma 2, we know that if fn⟶a.e.f with respect to Vg, then for any δ>0, there exists a closed set Fδ∈CT such that VgΩT\Fδ<δ and fn converges to f uniformly on Fδ.(3)
By Theorem1 (1) and Lemma 2, we know that if fn⟶a.e.f with respect to Vg, then there exists an increasing sequence of closed sets Hkk=1∞⊂ℱT such that(32)VgΩT\∪k=1∞Hk=0,and fn converges to f on Hk uniformly for any fixed k=1,2,….
In the following, we present Lusin’s theorem in the framework ofg-expectation.Theorem 2 (Lusin’s Theorem).
Suppose thatg satisfies (H2)–(H4) and f is an ℱT-measurable random variable. Then, for each δ>0, there exists a closed set Fδ∈CT such that VgΩT\Fδ<δ and f is continuous on Fδ.Proof.
We prove this theorem stepwise in the following two situations.(a)
Suppose thatf is a simple function, i.e., f=∑k=1nckχEk, where χEk is the characteristic function of Ek and ΩT=∪k=1nEk (a disjoint finite union). For any δ>0, by Lemma 2, we know that for each k, there exists a closed set Fk of ΩT such that Fk⊂Ek and(33)VgEk\Fk<δn.Let(34)Fδ≔∪k=1nFk.Then,Fδ is a closed set. By Remark 3, we have(35)VgΩT\Fδ=Vg∪k=1nEk\∪k=1nFk≤Vg∪k=1nEk\Fk≤∑k=1nVgEk\Fk<δ.Obviously,f is continuous on Fδ.(b)
Letf be an ℱT-measurable random variable. Then, there exists a sequence φnn=1∞ of simple functions such that φn⟶f on Ω, as n⟶∞. With the help of Remark 4 (3), we know that there exists an increasing sequence of closed sets Hkk=1∞⊂ℱT such that(36)VgΩT\∪k=1∞Hk=0,and φn converges to f on Hk uniformly for any fixed k=1,2,…. Applying (a), we can prove that for any fixed n, there exists a closed set Fkn of ΩT satisfying that Fkn⊂Hk such that(37)VgHk\Fkn<δ2n+k,k=1,2,…,and φn is continuous on Fkn. Let(38)Fδ≔∩k=1∞∩n=1∞Fkn.Then, Fδ is a closed set. By Remark 3, we have(39)VgΩT\Fδ=VgΩT\∪k=1∞Hk∪∪k=1∞Hk\∩k=1∞∩n=1∞Fkn≤VgΩT\∪k=1∞Hk+Vg∪k=1∞Hk\∩k=1∞∩n=1∞Fkn≤Vg∪n=1∞∪k=1∞Hk\Fkn≤∑n=1∞∑k=1∞VgHk\Fkn<δ.
At last, we show thatf is continuous on Fδ. In fact, φn is continuous and converges to f uniformly on Fδ. So for any ε>0 and any ω,ω0∈Fδ, there exist a positive integer no and a positive constant ς such that(40)φn0ω−fω<ε3,φn0ω−φn0ω0<ε3,when ω−ω0<ς. Thus, we have(41)fω−fω0=fω−φn0ω+φn0ω−φn0ω0+φn0ω0−fω0≤fω−φn0ω+φn0ω−φn0ω0+φn0ω0−fω0<ε.
Sof is continuous on Fδ.Remark 5.
Suppose thatg satisfies (H2)–(H4). By Theorem 2 and Lemma 2, we know that for any fixed n=1,2,…, there exists a closed sequence Fnn=1∞⊂ℱT such that f is continuous on Fn and(42)VgΩT\Fn<1n.
At last, we show continuous function approximation theorem in the framework ofg-expectation.Theorem 3 (Continuous Function Approximation Theorem).
Suppose thatg satisfies (H2)–(H4) and f is an ℱT-measurable random variable. Then, there exists a continuous function sequence ϕnn=1∞ on Ω such that ϕn⟶a.e.f with respect to Vg. Furthermore, if f≤M, then ϕn≤M n=1,2,…, where M is a positive constant.Proof.
By Remark5, we know that for every k=1,2,…, there exists a closed set Fk of ΩT such that f is continuous on Fk and VgΩT\Fk<1/k By Tietze’s extension theorem in Royden [27], for every k=1,2,…, there exists a continuous function ψk on Ω such that ψkω=fω, for ω∈Fk. And if f≤M, then ψk≤M. Therefore, for any ε>0, we have(43)ω:ψkω−fω≥ε⊂ΩT\Fk,And, hence, for any k=1,2,…,(44)Vgω:ψkω−fω≥ε≤VgΩT\Fk<1k.
Thus, we have(45)limk⟶∞Vgω:ψkω−fω≥ε=0.
From the above fact, we can choose a subsequenceψknn=1∞ of ψkk=1∞ such that(46)Vgω:ψknω−fω≥12n<12n.
Let(47)En≔ω:ψknω−fω≥12n.Then,(48)∑n=1∞VgEn<∞.
Next, we prove(49)Vg∩n=1∞∪v=1∞ω:ψkn+vω−fω≥ε=0.
Indeed, for anyε>0, there exists a positive integer n0 such that for any n≥n0, 1\2n<ε and(50)Vg∩n=1∞∪v=1∞ω:ψkn+vω−fω≥ε≤∑m=n∞Vgω:ψkmω−fω≥ε≤∑m=n∞VgEm⟶0,asn⟶∞.
That is,ψkn⟶a.e.f with respect to Vg, we take ϕn=ψkn, n=1,2,….



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