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Proceedings of the American Mathematical Society
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
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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?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0380968-7
PII: S 0002-9939(1974)0380968-7
Article copyright: © Copyright 1974 American Mathematical Society