Parametrizations of $G_{\delta }$-valued multifunctions

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 $F(t)$.