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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Parametrizations of $G_{\delta }$-valued multifunctions
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by H. Sarbadhikari and S. M. Srivastava PDF
Trans. Amer. Math. Soc. 258 (1980), 457-466 Request permission

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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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 258 (1980), 457-466
  • MSC: Primary 54C60; Secondary 54C65
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0558184-2
  • MathSciNet review: 558184