Egoroff’s Theorem and Lusin’s Theorem for Capacities in the Framework of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 9.93214 12.3952" width="9.93214pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-104" d="M546 430L539 434C529 434 505 438 495 440C473 444 450 448 430 448C352 448 265 412 213 366C145 306 96 203 96 103C96 22 135 -12 160 -12C190 -12 238 14 262 32C310 68 368 120 411 184H413C403 117 396 75 384 21C353 -118 325 -158 291 -184C270 -200 241 -205 208 -205C133 -205 90 -164 74 -110C70 -98 58 -100 49 -107C34 -119 23 -140 23 -155C23 -190 74 -261 166 -261C219 -261 280 -233 314 -208C383 -157 446 -79 470 81C491 223 529 388 546 430ZM456 386C452 357 433 283 420 252C402 216 366 174 325 129C288 88 239 56 212 56C192 56 182 77 182 120C182 165 199 242 226 292C256 348 281 377 311 389C327 395 353 402 375 402C408 402 436 394 456 386Z"/></g></svg>-</span>Expectation

Zong, Zhaojun; Hu, Feng; Tian, Xiaoxin

doi:https://doi.org/10.1155/2020/1450486

Mathematical Problems in Engineering

On this page

Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Stochastic Process Theory and Its Applications

View this Special Issue

Research Article | Open Access

Volume 2020 | Article ID 1450486 | https://doi.org/10.1155/2020/1450486

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

Zhaojun Zong,¹Feng Hu,^1,2and Xiaoxin Tian¹

Guest Editor: Wenguang Yu

Received26 Dec 2019

Accepted28 Jan 2020

Published20 Mar 2020

Abstract

In the classical real analysis theory, Egoroff’s theorem and Lusin’s theorem are two of the most important theorems. The 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 -expectation. We give some different assumptions that provide Egoroff’s theorem and Lusin’s theorem in the framework of -expectation.

1. Introduction

In the classical real analysis theory, Egoroff’s theorem and Lusin’s theorem are two of the most important theorems. The 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.

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 that was introduced by Dobrakov and Farkova [9].

The original motivation for studying nonlinear expectation and -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 -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 -expectation.

In this paper, we study Egoroff’s theorem and Lusin’s theorem for capacities induced by -expectation. We give the sufficient conditions that provide Egoroff’s theorem and Lusin’s theorem in the framework of -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 -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.

Let be a complete probability space and be a d-dimensional standard Brownian motion with respect to filtration generated by the Brownian motion and all P-null subsets, i.e.,where is the set of all P-null subsets. Fix a real number .

Let us introduce the following spaces:

Now, we consider the following 1-dimensional backward stochastic differential equation (BSDE):

Letsuch that for any , is -progressively measurable. We make the following assumptions: (H1) . (H2) There exists a constant such that for any and , (H3) For any and , . (H4) is subadditive with respect to y and z, i.e., for any and ,

Lemma 1 (see Pardoux and Peng [15]). Suppose that satisfies (H1) and (H2). Then, for any BSDE (2) has a unique pair of adapted processes .

Definition 1 (-expectation, see Peng [16]). Suppose that satisfies (H2) and (H3). For any , let be the solution of BSDE (2) with terminal value ξ. Consider the mapping , denoted by . We call the -expectation of ξ.
From Peng [16], we know that that -expectation keeps many properties of mathematical expectation:(i), if c is a constant(ii), if For more details of the properties of -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 satisfies (H2) and (H3). For any , there exists a positive constant C such that

Definition 2 (see Choquet [26]). A capacity is a real-valued set function satisfying(1)(2) whenever and Define the conjugate of by , Obviously, is also a capacity and .

Definition 3. Suppose that is a capacity. Then,(i)Countably subadditive:(ii)Continuity from above: for any , , whenever .(iii)Continuity from below: for any , , whenever .(iv)Continuity: is continuous from below and above.

Definition 4 (see Wang and Klir [1]). Let F be the class of all finite real-valued measurable functions on , and let :(i) converges almost everywhere to f on : there is a set such that and on (ii) converges pseudo almost everywhere to f on (): there is a set such that and on (iii) converges almost uniformly to f on (): for any , there is a set , such that and converges to f uniformly on (iv) converges to f pseudo almost uniformly on : there exists with such that converges to f on uniformly for any fixed

Remark 1. It is easy to prove that(1) with respect to if and only if with respect to (2) with respect to if and only if with respect to DefineIt is easy to check that is a capacity.

Remark 2. By Proposition 1, we can obtain that suppose satisfies (H2) and (H3), ; then(1), whenever (2), whenever Thus, is a continuous capacity. Similarly, is a continuous capacity.
The following proposition is a special case of Corollary 3.5 by Peng [12].

Proposition 2. Suppose that satisfies (H2)–(H4). Then,

Remark 3. Suppose that satisfies (H2)–(H4). By Remark 2 and Proposition 2, we haveThus, 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 of -expectation.

Theorem 1 (Egoroff’s Theorem). Suppose that satisfies (H2)–(H4), and f are -measurable random variables. Then,(1)If with respect to , then with respect to (2)If with respect to , then with respect to

Proof. Firstly, we prove Theorem 1 (1). Let D be the set of these points ω at which does not converge to f. Then,Since with respect to , we have . Thus, for any fixed positive integer k,Noting the fact thatand by Remark 2, we haveTherefore for any and any positive integer k, there exists a positive integer , such thatLetBy Remark 3, we haveThus, converges to f uniformly on . The proof of Theorem 1 (1) is complete.
From Theorem 1 (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: is the space of all -valued continuous paths with , equipped with the distanceWe set . It is clear that and are both complete separable metric spaces. Let and be the classes of open sets and closed sets in , respectively. Similarly, and are the classes of open sets and closed sets in , respectively.
We consider the canonical process: , for . Let be the smallest σ-algebra containing , and let be the smallest σ-algebra containing . We can choose a probability measure such that is a d-dimensional standard Brownian motion under .

Definition 5 (see Wu and Ha [5]). A capacity is called regular, if for every and , there exists a closed set and an open set of , such that

Lemma 2. Suppose that satisfies (H2)–(H4), then is regular on .

Proof. Let be the class of all sets such that for any , there exists a closed set and an open set of satisfyingTo prove this lemma, it is sufficient to show that .
Firstly, we verify that is an algebra. It is easy to know that . Suppose , then for any , there exist closed sets and open sets such thatSo we have is a closed set of , is an open set of , andThat is, . So is an algebra of .
Next, we prove that is closed under the formation of pairwise disjoint countable unions. Let be the sequence of pairwise disjoint set and be given. From the definition of and , we know that for each given n, there exist an open set and a closed set of such thatNoting the fact thatand by Remark 2, we haveThus, there exists a positive integer such thatDenote and ; then, is an open set of , is a closed set of , andBy Remark 3, we haveThat is,So is a σ-algebra of .
In real analysis theory, we know that for any closed set , there exists a sequence of open sets such thatTherefore, by Remark 2, we have Thus, Since is closed under the formation of complements, we have This shows that is a σ-algebra containing . So

Remark 4. Suppose that satisfies (H2)–(H4).(1)By Lemma 2, we know that for any , there exist an increasing sequence of closed sets and a decreasing sequence of open sets such that for every , (2)By Theorem 1 (1) and Lemma 2, we know that if with respect to , then for any , there exists a closed set such that and converges to f uniformly on .(3)By Theorem 1 (1) and Lemma 2, we know that if with respect to , then there exists an increasing sequence of closed sets such thatand converges to f on uniformly for any fixed .In the following, we present Lusin’s theorem in the framework of -expectation.

Theorem 2 (Lusin’s Theorem). Suppose that satisfies (H2)–(H4) and f is an -measurable random variable. Then, for each , there exists a closed set such that and f is continuous on .

Proof. We prove this theorem stepwise in the following two situations.(a)Suppose that f is a simple function, i.e., , where is the characteristic function of and (a disjoint finite union). For any , by Lemma 2, we know that for each k, there exists a closed set of such that and Let Then, is a closed set. By Remark 3, we have Obviously, f is continuous on .(b)Let f be an -measurable random variable. Then, there exists a sequence of simple functions such that on , as . With the help of Remark 4 (3), we know that there exists an increasing sequence of closed sets such thatand converges to f on uniformly for any fixed . Applying (a), we can prove that for any fixed n, there exists a closed set of satisfying that such thatand is continuous on . LetThen, is a closed set. By Remark 3, we haveAt last, we show that f is continuous on . In fact, is continuous and converges to f uniformly on . So for any and any , there exist a positive integer and a positive constant ς such thatwhen . Thus, we haveSo f is continuous on .

Remark 5. Suppose that satisfies (H2)–(H4). By Theorem 2 and Lemma 2, we know that for any fixed , there exists a closed sequence such that f is continuous on andAt last, we show continuous function approximation theorem in the framework of -expectation.

Theorem 3 (Continuous Function Approximation Theorem). Suppose that satisfies (H2)–(H4) and f is an -measurable random variable. Then, there exists a continuous function sequence on such that with respect to . Furthermore, if , then , where M is a positive constant.

Proof. By Remark 5, we know that for every , there exists a closed set of such that f is continuous on and By Tietze’s extension theorem in Royden [27], for every , there exists a continuous function on such that , for And if then . Therefore, for any , we haveAnd, hence, for any ,Thus, we haveFrom the above fact, we can choose a subsequence of such thatLetThen,Next, we proveIndeed, for any , there exists a positive integer such that for any , andThat is, with respect to , we take , .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported partly by the National Natural Science Foundation of China (no. 11801307) and the Natural Science Foundation of Shandong Province of China (nos. ZR2016JL002 and ZR2017MA012).

References

Z. Wang and G. J. Klir, Fuzzy Measure Theory, Plenum, New York, NY, USA, 1992.
J. Li, “A further investigation for Egoroffs theorem with respect to monotone set functions,” Kybernetika, vol. 39, no. 6, pp. 753–760, 2003.
View at: Google Scholar
J. Li and M. Yasuda, “Egoroff’s theorem on monotone non-additive measure spaces,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 12, no. 1, pp. 61–68, 2004.
View at: Publisher Site | Google Scholar
T. Murofushi, K. Uchino, and S. Asahina, “Conditions for Egoroff’s theorem in non-additive measure theory,” Fuzzy Sets and Systems, vol. 146, no. 1, pp. 135–146, 2004.
View at: Publisher Site | Google Scholar
C. Wu and M. Ha, “On the regularity of the fuzzy measure on metric fuzzy measure spaces,” Fuzzy Sets and Systems, vol. 66, no. 3, pp. 373–379, 1994.
View at: Publisher Site | Google Scholar
Q. Jiang and H. Suzuki, “Fuzzy measures on metric spaces,” Fuzzy Sets and Systems, vol. 83, no. 1, pp. 99–106, 1996.
View at: Publisher Site | Google Scholar
J. Kawabe, “Regularity and Lusin’s theorem for Riesz space-valued fuzzy measures,” Fuzzy Sets and Systems, vol. 158, no. 8, pp. 895–903, 2007.
View at: Publisher Site | Google Scholar
J. Li and R. Mesiar, “Lusin’s theorem on monotone measure spaces,” Fuzzy Sets and Systems, vol. 175, no. 1, pp. 75–86, 2011.
View at: Publisher Site | Google Scholar
I. Dobrakov and J. Farkova, “On submeasures II,” Mathematica Slovaca, vol. 30, pp. 65–81, 1980.
View at: Google Scholar
Z. Chen and L. Epstein, “Ambiguity, risk, and asset returns in continuous time,” Econometrica, vol. 70, no. 4, pp. 1403–1443, 2002.
View at: Publisher Site | Google Scholar
S. Peng, “Filtration consistent nonlinear expectations and evaluations of contingent claims,” Acta Mathematicae Applicatae Sinica, English Series, vol. 20, no. 2, pp. 191–214, 2004.
View at: Publisher Site | Google Scholar
S. Peng, “Modelling derivatives pricing mechanism with their generating functions,” 2006, http://arxiv.org/abs/math.PR/0605599v1.
View at: Google Scholar
E. Rosazza Gianin, “Risk measures via -expectations,” Insurance: Mathematics and Economics, vol. 39, no. 1, pp. 19–34, 2006.
View at: Publisher Site | Google Scholar
F. Hu, Z. Zong, and H. Wu, “Fubini theorem for non additive measures in the framework of -expectation,” Communications in Statistics—Theory and Methods, vol. 46, no. 23, pp. 11776–11785, 2017.
View at: Publisher Site | Google Scholar
E. Pardoux and S. G. Peng, “Adapted solution of a backward stochastic differential equation,” Systems & Control Letters, vol. 14, no. 1, pp. 55–61, 1990.
View at: Publisher Site | Google Scholar
S. Peng, “Backward SDE and related -expectation,” in Backward Stochastic Differential Equations, N. El Karoui and L. Mazliak, Eds., vol. 364, pp. 141–159, Longman, Harlow, UK, 1997, Pitman Research Notes in Mathematics Series.
View at: Google Scholar
P. Briand, F. Coquet, Y. Hu, J. Mémin, and S. Peng, “A converse comparison theorem for BSDEs and related properties of -expectation,” Electronic Communications in Probability, vol. 5, pp. 101–117, 2000.
View at: Publisher Site | Google Scholar
Z. Chen, T. Chen, and M. Davison, “Choquet expectation and Peng’s -expectation,” The Annals of Probability, vol. 33, no. 3, pp. 1179–1199, 2005.
View at: Publisher Site | Google Scholar
Z. Chen, R. Kulperger, and L. Jiang, “Jensens inequality for -expectation: part 1,” Comptes Rendus Mathematique, vol. 337, no. 11, pp. 725–730, 2003.
View at: Publisher Site | Google Scholar
L. Jiang, “Convexity, translation invariance and subadditivity for -expectations and related risk measures,” The Annals of Applied Probability, vol. 18, no. 1, pp. 245–258, 2008.
View at: Publisher Site | Google Scholar
K. He, M. Hu, and Z. Chen, “The relationship between risk measures and choquet expectations in the framework of -expectations,” Statistics & Probability Letters, vol. 79, no. 4, pp. 508–512, 2009.
View at: Publisher Site | Google Scholar
F. Hu, “Dynamically consistent nonlinear evaluations with their generating functions in ,” Acta Mathematica Sinica, English Series, vol. 29, no. 4, pp. 815–832, 2013.
View at: Publisher Site | Google Scholar
Z. Zong and F. Hu, “BSDEs under filtration-consistent nonlinear expectations and the corresponding decomposition theorem for ε-supermartingales in ,” Rocky Mountain Journal of Mathematics, vol. 43, no. 2, pp. 677–695, 2013.
View at: Publisher Site | Google Scholar
Z. Zong and F. Hu, “ weak convergence method on BSDEs with non-uniformly Lipschitz coefficients and its applications,” Proceedings of the Indian Academy of Sciences‐Mathematical Sciences, vol. 126, no. 3, pp. 433–460, 2016.
View at: Publisher Site | Google Scholar
Z. Zong, F. Hu, C. Yin, and H. Wu, “On Jensen’s inequality, Hölder’s inequality and Minkowski’s inequality for dynamically consistent nonlinear evaluations,” Journal of Inequalities and Applications, vol. 2015, no. 152, p. 152, 2015.
View at: Publisher Site | Google Scholar
G. Choquet, “Theory of capacities,” Annales de I’nstitut Fourier (Grenoble), vol. 5, pp. 131–295, 1953.
View at: Google Scholar
H. L. Royden, Real Analysis, Macmillan, New York, NY, USA, 1988.

Copyright

Copyright © 2020 Zhaojun Zong 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.

PDF Download Citation

Download other formats

Order printed copies

Views

1366

Downloads

762

Citations