Existence of a measurable function such that the pushforward of a measure is equal to another measure

I’m not familiar with your notations, but it seems that Theorem 17.41 from [Kech] might be useful for you.

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We recall some definitions used in this theorem.

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(17.4) A measure $\mu$ on a measurable space $X$ is called continuous if $\mu(\{x\})=0$ for all $x$.

(17.A) A measure $\mu$ on a measurable space $(X,\mathcal S)$ is a probability measure, if $\mu(X)=1$.

(17.E) Let $X$ be a separable metrizable space. We denote by $P(X)$ the set of probability Borel measures on $X$.

Let $X, Y$ be topological spaces. A map $f:X\to Y$ is Borel (measurable), if the inverse image of a Borel (equivalently: open or closed set is Borel). If $Y$ has a countable subbasis $\{V_n\}$, it is enough to require that $f^{-1}(V_n)$ is Borel for each $n$. We call $f$ a Borel isomorphism if it is a bijection and both $f$, $f^{-1}$ are Borel, i.e. for $A\subseteq X$, $A\in \mathbf{B}(X)\Leftrightarrow f(A)\in \mathbf{B}(Y)$.

References

[Kech] A. Kechris, Classical Descriptive Set Theory, – Springer, 1995.

Tags:

Probability Theory

Measure Theory

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