Sums, products and continuity of Borel maps in nonseparable metric spaces

Author:
R. W. Hansell

Journal:
Proc. Amer. Math. Soc. **104** (1988), 465-471

MSC:
Primary 28A05; Secondary 28A35, 28C10, 54H05

DOI:
https://doi.org/10.1090/S0002-9939-1988-0962814-3

MathSciNet review:
962814

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Abstract | References | Similar Articles | Additional Information

Abstract: We give a comparatively simple proof that the complex map will be (Borel measurable) of class , whenever and are of class , for not necessarily separable metric spaces and . The Borel measurability of other types of maps, such as the sum of two vector-valued maps, is easily deduced from this. A general version of this result, applicable to abstract measurable spaces, is also proven.

Our second principal result shows that, if is continuous in each variable separately, where and are metrizable, then has a -locally finite function base of closed sets, and thus will be continuous at each point of the complement of some first category subset of .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0962814-3

Keywords:
Borel measurable maps,
nonseparable metric spaces,
complex maps,
separately continuous maps

Article copyright:
© Copyright 1988
American Mathematical Society