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Proceedings of the American Mathematical Society

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

 

 

Liftings of functions with values in a completely regular space


Authors: G. A. Edgar and Michel Talagrand
Journal: Proc. Amer. Math. Soc. 78 (1980), 345-349
MSC: Primary 46G15; Secondary 28A51
DOI: https://doi.org/10.1090/S0002-9939-1980-0553373-0
MathSciNet review: 553373
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Abstract: Let T be a completely regular space, let $ (\Omega ,\mathcal{F},\mu )$ be complete probability space, and let $ \rho :{\mathcal{L}^\infty }(\mu ) \to {\mathcal{L}^\infty }(\mu )$ be a lifting. If $ f:\Omega \to T$ is a Baire measurable function, must there exist a function $ \tilde f$ with almost all of its values in T, such that $ \rho (h \circ f) = h \circ \tilde f$ for all bounded continuous functions h on T? If T is strongly measure-compact, then the answer is ``yes". If T is not measure-compact, then the answer is ``no". This shows that a lifting is not always the best method for the construction of weak densities for vector measures.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0553373-0
Article copyright: © Copyright 1980 American Mathematical Society