- Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space.
- Let $$ X, Y : \Omega \rightarrow \mathbb{R} $$ be random variables.
- Furthermore, let

$$ f: \mathbb{R}^2 \rightarrow \mathbb{R} $$ be a $\mathcal{B}(\mathbb{R}^2)/\mathcal{B}(\mathbb{R})$-measurable function such that, for all $y \in \mathbb{R}$, the random variables $f(X,y)$ and $f(X,Y)$ have finite expectation.

Now let $y \in \mathbb{R}$ be arbitrary. Under the above assumptions, the expected value $\mathbb{E}[f(X,y)]$ and a $\mathbb{P}$-unique conditional expectation $\mathbb{E}[f(X,Y) \mid Y]$ do exist.

Furthermore, since $\mathbb{E}[f(X,Y) \mid Y]$ is $\sigma(Y)/\mathcal{B}(\mathbb{R})$-measurable, there exists a $\mathbb{P}_Y$-unique $\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R})$-measurable function $$ \varphi : \mathbb{R} \rightarrow \mathbb{R} $$ such that $\varphi(Y) = \mathbb{E}[f(X,Y) \mid Y]$.

Under which circumstances does it hold, that $\varphi$ can be chosen such that $$ \varphi (y) = \mathbb{E}[f(X,y)] $$ and why?

Thanks in advance for any advice!

regular conditional probabilit(sometimes called atransition kernel) is usually helpful with this sort of thing. Do you have a particular set of circumstances where you'd like your $\varphi$ to exist? $\endgroup$