Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Weak convergences of probability measures:
A uniform principle


Author: Jean B. Lasserre
Journal: Proc. Amer. Math. Soc. 126 (1998), 3089-3096
MSC (1991): Primary 60B05, 60B10, 28A33
DOI: https://doi.org/10.1090/S0002-9939-98-04390-1
MathSciNet review: 1452809
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a set $\prod$ of probability measures on a locally compact separable metric space. It is shown that a necessary and sufficient condition for (relative) sequential compactness of $\prod$ in various weak topologies (among which the vague, weak and setwise topologies) has the same simple form; i.e. a uniform principle has to hold in $\prod$. We also extend this uniform principle to some Köthe function spaces.


References [Enhancements On Off] (What's this?)

  • 1. R.B. Ash, Real Analysis and Probability, Academic Press, New York, 1972. MR 55:8280
  • 2. E.J. Balder, On compactness of the space of policies in stochastic dynamic programming, Stoch. Proc. Appl. 32, pp. 141-150, 1989. MR 91b:90212
  • 3. J. Dieudonné, Sur la convergence des suites de mesures de Radon, Anais Acad. Brasil. Ci. 23, pp. 21-38, 1951. MR 13:121a
  • 4. A.G. Bhatt, V. Borkar, Occupation measures for controlled Markov processes: characterization and optimality, Ann. Prob. 24 (1996), 1531-1562. MR 97i:90105
  • 5. J.L. Doob, Measure Theory, Springer-Verlag, New York, 1994. MR 95c:28001
  • 6. N. Dunford, J.T. Schwartz, Linear Operators. Part I: General Theory, Interscience Publishers, Inc., New York, 1957. MR 90g:47001a
  • 7. W.H. Fleming, D. Vermes, Convex duality approach to the optimal control of diffusions, SIAM J. Contr. Optim. 27, pp. 1136-1155, 1989. MR 90i:49016
  • 8. O. Hernandez-Lerma, J.B. Lasserre, Discrete-Time Markov Control Processes: Basic Optimality Criteria, Springer-Verlag, New York, 1996. MR 96k:93001
  • 9. M. Kurano, M. Kawai, Existence of optimal stationary policies in discounted decision processes, Comp. Math. Appl. 27, pp. 95-101, 1994. MR 95a:90197
  • 10. J. Lindenstrauss, L. Tzafriri, Classical Banach spaces I and II, Springer-Verlag, Berlin, 1977. MR 81c:46001; MR 58:17766
  • 11. E. Mascolo, L. Migliaccio, Relaxation methods in control theory, Appl. Math. Optim. 20, pp. 97-103, 1989. MR 90c:49059
  • 12. J.E. Rubio, The global control of nonlinear diffusion equations, SIAM J. Contr. Optim. 33, pp. 308-322, 1995. MR 90b:49009
  • 13. K. Yosida, Functional Analysis, Springer-Verlag, New York, 1980. MR 82i:46002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 60B05, 60B10, 28A33

Retrieve articles in all journals with MSC (1991): 60B05, 60B10, 28A33


Additional Information

Jean B. Lasserre
Affiliation: LAAS-CNRS, 7 Av. du Colonel Roche, 31077 Toulouse Cédex, France
Email: lasserre@laas.fr

DOI: https://doi.org/10.1090/S0002-9939-98-04390-1
Keywords: Sequences of measures, weak convergences of measures, Banach lattices, K\"{o}the function spaces
Received by editor(s): November 25, 1996
Received by editor(s) in revised form: March 10, 1997
Communicated by: Stanley Sawyer
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society