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Let $X$ and $Y$ be two Polish spaces, and $\mu$ and $\nu$ be Borel probability measures on $X$ and $Y$, respectively. Why is there no Borel map $T:X \to Y$ such that $$ \nu(E) = \mu (T^{-1}(E)) $$ when $\mu$ is Dirac delta and $\nu$ is not?

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Suppose we have such a Borel map $T:X\to Y$ and let $\mu$ be a Dirac measure at some $x\in X$, i.e. $\mu=\delta_{x}$, whence $\nu(E)=\delta_{x}(T^{-1}(E))$ for all Borel sets $E$. In other words we have $\nu(E)=1$ if $x\in T^{-1}(E)$, or equivalently if $T(x)\in E$, and $\nu(E)=0$ otherwise, which implies that $\nu=\delta_{T(x)}$.

Hence it is impossible for $\nu$ not be a dirac measure and such $T$ to exist.

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