In this paper we prove two convergence theorems for set-valued conditional expectations. The first is a set-valued generalization of Levy's martingale convergence theorem, while the second involves a nonmonotone sequence of sub $\sigma $-fields.
@article{118560,
author = {Nikolaos S. Papageorgiou},
title = {Convergence theorems for set-valued conditional expectations},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {34},
year = {1993},
pages = {97-104},
zbl = {0788.60021},
mrnumber = {1240208},
language = {en},
url = {http://dml.mathdoc.fr/item/118560}
}
Papageorgiou, Nikolaos S. Convergence theorems for set-valued conditional expectations. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 97-104. http://gdmltest.u-ga.fr/item/118560/
The optional sampling theorem for convex set valued martingales, J. Reine Angew. Math. 310 (1979), 1-6. (1979) | MR 0546661
Law of large numbers for random sets and allocation processes, Math. Oper. Res. 6 (1981), 485-492. (1981) | MR 0703091 | Zbl 0524.28015
Famille d'opérateurs maximaux monotones et mesurabilité, Ann. Mat. Pura ed Appl. 120 (1979), 35-111. (1979) | MR 0551062
Vector Measures, Math. Surveys, vol. 15, AMS, Providence, RI, 1977. | MR 0453964
Regular conditional expectations of correspondences, Theory of Prob. and Appl. 21 (1976), 325-338. (1976) | MR 0430204 | Zbl 0367.60002
On the continuity of conditional expectations, J. Math. Anal. Appl. 61 (1977), 227-231. (1977) | MR 0455110 | Zbl 0415.60003
Atomes conditionels d'un espace de probabilité, Acta Math. Hungarica 17 (1966), 443-449. (1966) | MR 0205285
Measurability and integrability of the weak upper limit of a sequence of multifunctions, J. Math. Anal. Appl. 153 (1990), 206-249. (1990) | MR 1080128 | Zbl 0748.47046
Radon-Nikodym theorems for set-valued measures, J. Multiv. Anal. 8 (1978), 96-118. (1978) | MR 0583862 | Zbl 0384.28006
Integrals, conditional expectations and martingales of multivalued functions, J. Multiv. Anal. 7 (1977), 149-182. (1977) | MR 0507504 | Zbl 0368.60006
A convergence theorem for convex set-valued supermartingales, Stoch. Anal. Appl. 3 (1985), 433-445. (1985) | MR 0808943
Quelques resultats de representation des amarts uniforms multivoques, C.R. Acad. Su. Paris 300 (1985), 63-63. (1985)
Semimartingales, DeGruyter, Berlin 1982. | MR 0688144 | Zbl 0595.60008
Convergence of convex sets and solutions of variational inequalities, Advances in Math. 3 (1969), 510-585. (1969) | MR 0298508
On the efficiency and optimality of allocations II, SIAM J. Control Optim. 24 (1986), 452-479. (1986) | MR 0838050 | Zbl 0589.90015
Convergence theorem for Banach space valued integrable multifunctions, Intern. J. Math. and Math. Sci. 10 (1987), 433-442. (1987) | MR 0896595
On the theory of Banach space valued multifunctions. Part 1: Integration and conditional expectation, J. Multiv. Anal. 17 (1985), 185-206. (1985) | MR 0808276
On the theory of Banach space valued multifunctions. Part 2: Set valued martingales and set valued measures, J. Multiv. Anal. 17 (1985), 207-227. (1985) | MR 0808277
A convergence theorem for set-valued supermartingales in a separable Banach space, Stoch. Anal. Appl. 5 (1988), 405-422. (1988) | MR 0912867
Convergence in approximation and nonsmooth analysis, J. Approx. Theory 49 (1987), 41-54. (1987) | MR 0870548 | Zbl 0619.41033
On the convergence of sequences of convex sets in finite dimensions, SIAM Review 21 (1979), 18-33. (1979) | MR 0516381
Esperances conditionelles d'integrandes semicontinus, Ann. Inst. H. Poincaré Ser. B 17 (1981), 337-350. (1981) | MR 0644351
Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), 859-903. (1977) | MR 0486391 | Zbl 0407.28006