Egoroff’s Theorem and Lusin’s Theorem for Capacities in the Framework of g-Expectation

In the classical real analysis theory, Egoroff’s theorem and Lusin’s theorem are two of the most important theorems. /e σ-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 gexpectation.


Introduction
In the classical real analysis theory, Egoroff's theorem and Lusin's theorem are two of the most important theorems. e σ-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. e 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]. ese 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 spacevalued 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]. e original motivation for studying nonlinear expectation and g-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 by g-expectation. We give the sufficient conditions that provide Egoroff's theorem and Lusin's theorem in the framework of g-expectation. e 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.

Preliminaries
In this section, we shall present some notations, assumptions, notions, lemmas, and propositions that are used in this paper.
Definition 4 (see Wang and Klir [1]). Let F be the class of all finite real-valued measurable functions on (Ω, F, V), and let f, f n ∈ F (n � 1, 2, . . .): It is easy to check that V g is a capacity.

Main Results
In this section, we study Egoroff's theorem, Lusin's theorem, and continuous function approximation theorem in the framework of g-expectation.  Proof. Firstly, we prove eorem 1 (1). Let D be the set of these points ω at which f n does not converge to f. en, Since f n ⟶ a.e. f with respect to V g , we have V g (D) � 0.
us, for any fixed positive integer k, Noting the fact that and by Remark 2, we have erefore for any δ > 0 and any positive integer k, there exists a positive integer N k , such that Let By Remark 3, we have (16) us, f n converges to f uniformly on E δ . e proof of eorem 1 (1) is complete. From eorem 1 (1) and by Remark 1, we can easily obtain eorem 1 (2).

Mathematical Problems in Engineering
We consider the canonical process: ω t � W t (ω), t ∈ [0, ∞), for ω ∈ Ω. Let F be the smallest σ-algebra containing O, and let F T be the smallest σ-algebra containing O T . We can choose a probability measure P such that (W t ) t≥0 is a d-dimensional standard Brownian motion under (C d 0 (R + ), F, P). □ Definition 5 (see Wu and Ha [5]). A capacity V is called regular, if for every A ∈ F and δ > 0, there exists a closed set F δ and an open set G δ of Ω, such that Lemma 2. Suppose that g satisfies (H2)-(H4), then V g is regular on F T .
Proof. Let A be the class of all sets A ∈ F T such that for any δ > 0, there exists a closed set F δ and an open set G δ of Ω T satisfying To prove this lemma, it is sufficient to show that F T ⊂ A. Firstly, we verify that A is an algebra. It is easy to know that Ω T ∈ A. Suppose A, B ∈ A, then for any δ > 0, there exist closed sets F 1,δ , F 2,δ ∈ Ω T and open sets G 1,δ , G 2,δ ∈ Ω T such that So we have δ is an open set of Ω T , and (22) at is, A\B ⊂ A. So A is an algebra of Ω T . Next, we prove that A is closed under the formation of pairwise disjoint countable unions. Let A n ∞ n�1 ⊂ A be the sequence of pairwise disjoint set and δ > 0 be given. From the definition of A and A n ∈ A, we know that for each given n, there exist an open set G n and a closed set F n of Ω T such that F n ⊂ A n ⊂ G n , V g G n \F n < δ 2 n+1 . (23) Noting the fact that and by Remark 2, we have us, there exists a positive integer k 0 such that Denote G δ : � ∪ ∞ n�1 G n and F δ : � ∪ k 0 n�1 F n ; then, G δ is an open set of Ω T , F δ is a closed set of Ω T , and By Remark 3, we have at is, So A is a σ-algebra of Ω T . In real analysis theory, we know that for any closed set F ∈ C T , there exists a sequence of open sets E n ∞ n�1 such that E n \F↘∅, as n ⟶ ∞.
(30) erefore, by Remark 2, we have lim n⟶∞ V g (E n \F) � 0. us, C T ⊂ A. Since A is closed under the formation of complements, we have O T ⊂ A.
is shows that A is a σ-algebra containing O T . So F T ⊂ A. □ Remark 4. Suppose that g satisfies (H2)-(H4).
(1) By Lemma 2, we know that for any A ∈ F T , there exist an increasing sequence F n ∞ n�1 of closed sets and a decreasing sequence G n ∞ n�1 of open sets such that for every n � 1, 2, . . ., F n ⊂ A ⊂ G m , V g G n \A < 1 n , V g A\F n < 1 n . (31) (2) By eorem 1 (1) and Lemma 2, we know that if f n ⟶ a.e. f with respect to V g , then for any δ > 0, there exists a closed set F δ ∈ C T such that V g (Ω T \F δ ) < δ and f n converges to f uniformly on F δ . (3) By eorem 1 (1) and Lemma 2, we know that if f n ⟶ a.e. f with respect to V g , then there exists an and f n converges to f on H k uniformly for any fixed k � 1, 2, . . ..
In the following, we present Lusin's theorem in the framework of g-expectation.
Theorem 2 (Lusin's eorem). Suppose that g satisfies (H2)-(H4) and f is an F T -measurable random variable. en, for each δ > 0, there exists a closed set F δ ∈ C T such that V g (Ω T \F δ ) < δ and f is continuous on F δ .
Proof. We prove this theorem stepwise in the following two situations.
(a) Suppose that f is a simple function, i.e., f � n k�1 c k χ E k , where χ E k is the characteristic function of E k and Ω T � ∪ n k�1 E k (a disjoint finite union). For any δ > 0, by Lemma 2, we know that for each k, there exists a closed set F k of Ω T such that F k ⊂ E k and Let en, F δ is a closed set. By Remark 3, we have Obviously, f is continuous on F δ .
(b) Let f be an F T -measurable random variable. en, there exists a sequence φ n ∞ n�1 of simple functions such that φ n ⟶ f on Ω, as n ⟶ ∞. With the help of Remark 4 (3), we know that there exists an increasing sequence of closed sets H k ∞ k�1 ⊂ F T such that and φ n converges to f on H k uniformly for any fixed k � 1, 2, . . .. Applying (a), we can prove that for any fixed n, there exists a closed set and φ n is continuous on en, F δ is a closed set. By Remark 3, we have At last, we show that f is continuous on F δ . In fact, φ n is continuous and converges to f uniformly on F δ . So for any ε > 0 and any ω, ω 0 ∈ F δ , there exist a positive integer n o and a positive constant ς such that when |ω − ω 0 | < ς. us, we have So f is continuous on F δ .

Mathematical Problems in Engineering
Remark 5. Suppose that g satisfies (H2)-(H4). By eorem 2 and Lemma 2, we know that for any fixed n � 1, 2, . . ., there exists a closed sequence F n ∞ n�1 ⊂ F T such that f is continuous on F n and At last, we show continuous function approximation theorem in the framework of g-expectation. Proof. By Remark 5, we know that for every k � 1, 2, . . ., there exists a closed set F k of Ω T such that f is continuous on F k and V g (Ω T \F k ) < (1/k) By Tietze's extension theorem in Royden [27], for every k � 1, 2, . . ., there exists a continuous function ψ k on Ω such that ψ k (ω) � f(ω), for ω ∈ F k . And if |f| ≤ M, then |ψ k | ≤ M. erefore, for any ε > 0, we have

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.