## 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