On the convergence of closed-valued measurable multifunctions

Authors:
Gabriella Salinetti and Roger J.-B. Wets

Journal:
Trans. Amer. Math. Soc. **266** (1981), 275-289

MSC:
Primary 28A20; Secondary 54C60

MathSciNet review:
613796

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the convergence almost everywhere and in measure of sequences of closed-valued multifunctions. We first give a number of criteria for the convergence of sequences of closed subsets. These results are used to obtain various characterizations for the convergence of measurable multifunctions. In particular we are interested in the convergence properties of (measurable) selections.

**[1]**C. Castaing and M. Valadier,*Convex analysis and measurable multifunctions*, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR**0467310****[2]**R. Tyrrell Rockafellar,*Integral functionals, normal integrands and measurable selections*, Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975) Springer, Berlin, 1976, pp. 157–207. Lecture Notes in Math., Vol. 543. MR**0512209****[3]**G. Choquet,*Convergences*, Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.)**23**(1948), 57–112. MR**0025716****[4]**Ernest Michael,*Topologies on spaces of subsets*, Trans. Amer. Math. Soc.**71**(1951), 152–182. MR**0042109**, 10.1090/S0002-9947-1951-0042109-4**[5]**D. W. Curtis and R. M. Schori,*2^{𝑥} and 𝐶(𝑋) are homeomorphic to the Hilbert cube*, Bull. Amer. Math. Soc.**80**(1974), 927–931. MR**0353235**, 10.1090/S0002-9904-1974-13579-2**[6]**Gerard Debreu,*Integration of correspondences*, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 351–372. MR**0228252****[7]**J. Neveu,*Convergence presque sûre de martingales multivoques*, Ann. Inst. H. Poincaré Sect. B (N.S.)**8**(1972), 1–7 (French, with English summary). MR**0331504****[8]**G. Matheron,*Random sets and integral geometry*, John Wiley & Sons, New York-London-Sydney, 1975. With a foreword by Geoffrey S. Watson; Wiley Series in Probability and Mathematical Statistics. MR**0385969****[9]**Gabriella Salinetti and Roger J.-B. Wets,*On the convergence of sequences of convex sets in finite dimensions*, SIAM Rev.**21**(1979), no. 1, 18–33. MR**516381**, 10.1137/1021002**[10]**Maurice Sion,*A theory of semigroup valued measures*, Lecture Notes in Mathematics, Vol. 355, Springer-Verlag, Berlin-New York, 1973. MR**0450503**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
28A20,
54C60

Retrieve articles in all journals with MSC: 28A20, 54C60

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1981-0613796-3

Keywords:
Measurable multifunction,
measurable selection,
convergence in probability,
convergence almost everywhere,
hyperspaces,
convergence in hyperspaces

Article copyright:
© Copyright 1981
American Mathematical Society