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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Pointwise in terms of weak convergence
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by J. R. Baxter PDF
Proc. Amer. Math. Soc. 46 (1974), 395-398 Request permission


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.
  • J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
  • Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
  • 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.
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 46 (1974), 395-398
  • MSC: Primary 60G45
  • DOI:
  • MathSciNet review: 0380968