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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Representation of functions as limits of martingales


Author: Charles W. Lamb
Journal: Trans. Amer. Math. Soc. 188 (1974), 395-405
MSC: Primary 60G45
MathSciNet review: 0339328
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Abstract: In this paper we show that if $ (\Omega ,\mathcal{F},P)$ is a probability space and if $ {\{ \mathcal{F}{_n}\} _{n \geq 1}}$ is an increasing sequence of sub-$ \sigma $-fields of $ \mathcal{F}$ which satisfy an additional condition, then every real valued, $ {\mathcal{F}_\infty }$-measurable function f can be written as the a.e. limit of a martingale $ {\{ {f_n},{\mathcal{F}_n}\} _{n \geq 1}}$. The case where f takes values in the extended real line is also studied. A construction is given of a ``universal'' martingale $ {\{ {f_n},{\mathcal{F}_n}\} _{n \geq 1}}$ such that any $ {\mathcal{F}_\infty }$-measurable function is the a.e. limit of a suitably chosen subsequence.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0339328-1
PII: S 0002-9947(1974)0339328-1
Keywords: Martingales, H-systems, universal martingales
Article copyright: © Copyright 1974 American Mathematical Society