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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Parametrizations of $ G\sb{\delta }$-valued multifunctions


Authors: H. Sarbadhikari and S. M. Srivastava
Journal: Trans. Amer. Math. Soc. 258 (1980), 457-466
MSC: Primary 54C60; Secondary 54C65
MathSciNet review: 558184
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Abstract: Let T, X be Polish spaces, $ \mathcal{J}$ a countably generated sub-$ \sigma $-field of $ {\mathcal{B}_T}$, the Borel $ \sigma $-field of T, and $ F:\,T\, \to \,X$ a multifunction such that $ F(t)$ is a $ {G_\delta }$ in X for each $ t\, \in \,T$. F is $ \mathcal{J}$-measurable and $ {\text{Gr}}(F)\, \in \,J\, \otimes \,{\mathcal{B}_X}$, where $ {\text{Gr}}(F)$ denotes the graph of F. We prove the following three results on F.

(I) There is a map $ f:\,T\, \times \,\Sigma \, \to \,X$ such that for each $ t\, \in \,T,\,f(t,\, \cdot )$ is a continuous, open map from $ \Sigma $ onto $ F(t)$ and for each $ \sigma \, \in \,\Sigma ,\,f( \cdot ,\,\sigma )$ is $ \mathcal{J}$-measurable, where $ \Sigma $ is the space of irrationals.

(II) The multifunction F is of Souslin type.

(III) If X is uncountable and $ F(t),\,t\, \in \,T$, are all dense-in-itself then there is a $ \mathcal{J}\, \otimes \,{\mathcal{B} _X}$-measurable map $ f:\,T\, \times \,X\, \to \,X$ such that for each $ t\, \in \,T,\,f(t,\, \cdot )$ is a Borel isomorphism of X onto $ F(t)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0558184-2
PII: S 0002-9947(1980)0558184-2
Keywords: Multifunctions, selectors, parametrizations, representations, uniformizations
Article copyright: © Copyright 1980 American Mathematical Society