If $(Ω,A,text P)$ is a non-atomic prob. space and $μ$ is a prob. measure on a Borel space there is a...
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Let $(Omega,mathcal A,operatorname P)$ be a non-atomic probability space $(E,mathcal E)$ be a Borel space $mu$ be a probability measure on $(E,mathcal E)$ How can we show that there is a $(mathcal A,mathcal E)$ -measurable $X:Omegato E$ with $X_astoperatorname P=mu$ (pushforward measure)? I know that if $(mathbb R,mathcal B(mathbb R),nu)$ is a non-atomic probability space, the distribution function $F$ of $nu$ is continous and $F_astnu=mathcal U_{[0,:1]}$ (uniform distribution). Now, I'll assume that for all $Binmathcal B(mathbb R)$ , $(B,mathcal B(B))$ is isomorphic to $(mathbb R,mathcal B(mathbb R))$ (i.e. there is a measurable bijection between $(B,mathcal B(B))$ and $(mathbb R,mathcal B(mathbb R))$ with measurable inverse). Does anyone have a reference for t
