Liftings of functions with values in a completely regular space
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- by G. A. Edgar and Michel Talagrand
- Proc. Amer. Math. Soc. 78 (1980), 345-349
- DOI: https://doi.org/10.1090/S0002-9939-1980-0553373-0
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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.References
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Bibliographic Information
- © Copyright 1980 American Mathematical Society
- 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