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)



Pointwise in terms of weak convergence

Author: J. R. Baxter
Journal: Proc. Amer. Math. Soc. 46 (1974), 395-398
MSC: Primary 60G45
MathSciNet review: 0380968
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (\Omega ,\mathfrak{F},\mu )$ be a measure space, $ \mu (\Omega ) < \infty $. Let $ {X_n}$ be a sequence of measurable functions on $ \Omega $ taking values in a compact metric space $ M$. The set of bounded stopping times $ \tau $ for the $ {X_n}$ is a directed set under the obvious ordering. The following theorem is proved: $ {X_n}$ converges pointwise almost everywhere if and only if the generalized sequence $ \int {\phi ({X_\tau })d\mu } $ converges for every continuous function $ \phi $ on $ M$. The martingale theorem is proved as an application.

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

  • [1] J. L. Doob, Stochastic processes, Wiley, New York; Chapman & Hall, London, 1953. MR 15, 445. MR 0058896 (15:445b)
  • [2] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [3] P. A. Meyer, Le retournement du temps, d'apres Chung et Walsh, Séminaire de Probabilités V, Univ. de Strasbourg, Lecture Notes in Math., vol. 191, Springer-Verlag, New York, 1971.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60G45

Retrieve articles in all journals with MSC: 60G45

Additional Information

Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society