Let $X, Y$ be Polish spaces and $\mu, \nu$ be Borel probablity measures on $X, Y$, respectively. Assume that $\mu$ is non-atomic. Denote by $\mathcal{M}(\mu,\nu)$ a set of Borel measurable mappings $T$ from $X$ to $Y$ which preserve measures (i.e. for any Borel set $B$ of $Y$, we have $\mu(T^{-1}(B))=\nu(B)$). Is the set $\mathcal{M}(\mu,\nu)$ compact with respect to the convergence of $\mu$-measure?

  • $\begingroup$ What do mean with 'with respect to the convergence of $\mu$-measure'? $\endgroup$ – p4sch Dec 7 '18 at 12:29
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    $\begingroup$ I made some mistakes. Convergence ''in'' $\mu$-measure, not convergence ''of'' $\mu$-measure. We need to fix a distance $d$ on $Y$. A sequence ${T_n}$ of Borel measurable maps from $X$ to $Y$ converges in $\mu$-measure to a Borel measurable map $T\colon X\to Y$ if for any $\epsilon>0$, $\mu(\{x\in X \mid d(T_n(x),T(x))>\varepsilon\})$ tends to $0$ as $n$ tends to infinity. $\endgroup$ – Nick Dec 7 '18 at 13:04

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