$(\Omega,\mathcal{F},P),(\Omega',\mathcal{F}')$ are two probability spaces, $X:\Omega \to \Omega'$ is $(\mathcal{F},\mathcal{F}')$-measurable, $P':=P_X,E \in \mathcal{F}.$
It's very easy to verify the following statement:
- for any $\sigma$-algebra $\mathcal{G}' \subset \mathcal{F}',E' \in \mathcal{F}',P(X^{-1}(E')|X^{-1}(\mathcal{G'}))(w)=P'(E'|\mathcal{G}')(X(w))$ for $P$-almost every $w \in \Omega.$ ($X^{-1}(\mathcal{G}')$ the smallest $\sigma$-algebra on $\Omega'$ which makes $X$ measurable)
- there exists $f: \Omega' \to \mathbb{R}_+,\mathcal{F}'$-measurable such that for all $E' \in \mathcal{F}',P(E \cap X^{-1}(E'))=\int_{E'}f dP'$ (density), so $foX$ is a version of $P(E|X^{-1}(\mathcal{F}')).$
Is there any relation between statements 1. and 2.?