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Proceedings of the American Mathematical Society

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



A note on some properties of $ A$-functions

Author: H. Sarbadhikari
Journal: Proc. Amer. Math. Soc. 56 (1976), 321-324
MSC: Primary 26A42
MathSciNet review: 0407213
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Abstract: This note deals with $ ({\mathbf{M}},\ast)$ functions for various families $ {\mathbf{M}}$. It is shown that if $ {\mathbf{M}}$ is the family of Borel sets of additive class $ \alpha $ on a metric space $ X$, then $ ({\mathbf{M}},\ast)$ functions are just the functions of the form $ {\sup _y}g(x,y)$ where $ g:X \times R \to R$ is continuous in $ y$ and of class $ \alpha $ in $ x$. If $ {\mathbf{M}}$ is the class of analytic sets in a Polish space $ X$, then the $ ({\mathbf{M}},\ast)$ functions dominating a Borel function are just the functions $ {\sup _y}g(x,y)$ where $ g$ is a real valued Borel function on $ {X^2}$. It is also shown that there is an $ A$-function $ f$ defined on an uncountable Polish space $ X$ and an analytic subset $ C$ of the real line such that $ {f^{ - 1}}(C) \notin $ the $ \sigma $-algebra generated by the analytic sets on $ X$.

References [Enhancements On Off] (What's this?)

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Keywords: $ ({\mathbf{M}},\ast)$ functions, $ A$-functions, $ \alpha $-functions, complete ordinary function system, functions of class $ \alpha $, operation $ A$
Article copyright: © Copyright 1976 American Mathematical Society