The Carathéodory extension theorem for vector valued measures

Kupka, Joseph

doi:10.1090/S0002-9939-1978-0493327-7

The Carathéodory extension theorem for vector valued measures
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by Joseph Kupka

Proc. Amer. Math. Soc. 72 (1978), 57-61

DOI: https://doi.org/10.1090/S0002-9939-1978-0493327-7

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Abstract:

This paper comprises three advertisements for a known theorem which, the author believes, deserves the title of the Carathéodory extension theorem for vector valued premeasures. Principal among these is a short and transparent proof of Porcelli’s criterion for the weak convergence of a sequence in the Banach space of bounded finitely additive complex measures defined on an arbitrary field, and equipped with the total variation norm. Also, a characterization of the so-called Carathéodory Extension Property is presented, and there is a brief discussion of the relevance of this material to stochastic integration.

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