Representation of functions as limits of martingales

Lamb, Charles W.

doi:10.1090/S0002-9947-1974-0339328-1

Representation of functions as limits of martingales
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by Charles W. Lamb

Trans. Amer. Math. Soc. 188 (1974), 395-405

DOI: https://doi.org/10.1090/S0002-9947-1974-0339328-1

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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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Sur les nombres dérivés

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