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  • 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!

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  • $\begingroup$ The notion of a regular conditional probabilit (sometimes called a transition 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? $\endgroup$ – DCM Apr 12 at 17:40
  • $\begingroup$ *probability, not 'probabilit'... $\endgroup$ – DCM Apr 12 at 17:50
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$\newcommand{\R}{\mathbb{R}} \newcommand{\vpi}{\varphi}$ The answer is: the condition

$E(f(X,Y)|Y)=\vpi(Y)$ for all $f$ such that $f(X,Y)$ has a finite expectation

holds iff $X$ and $Y$ are independent.

Indeed, if $X$ and $Y$ are independent, then for each Borel subset $B$ of $\R$ \begin{align*} EI\{Y\in B\}\vpi(Y)&=\int P(Y\in dy)I\{y\in B\}\vpi(y) \\ &=\int P(Y\in dy)I\{y\in B\}Ef(X,y) \\ &=\int P(Y\in dy)I\{y\in B\}\int P(X\in dx)f(x,y) \\ &=\int P(X\in dx,Y\in dy)I\{y\in B\}f(x,y) \\ &=EI\{Y\in B\}f(X,Y), \end{align*} where $I$ is the indicator. So, $E(f(X,Y)|Y)=\vpi(Y)$.

Vice versa, suppose that $E(f(X,Y)|Y)=\vpi(Y)$ for all $f$ such that $f(X,Y)$ has a finite expectation, where $\vpi(Y)=Ef(X,y)$ for all real $y$. Take any Borel subsets $A$ and $B$ of $\R$, and let $f(x,y):=I\{x\in A\}I\{y\in B\}$ for all real $x,y$. Then for each $y\in\R$ \begin{equation} \vpi(y)=Ef(X,y)=EI\{X\in A\}I\{y\in B\}=P(X\in A)I\{y\in B\} \end{equation} and \begin{align*} P(X\in A)P(Y\in B)&=EP(X\in A)I\{Y\in B\} \\ &=EI\{Y\in B\}\vpi(Y) \\ &=EI\{Y\in B\}f(X,Y) \\ &=EI\{X\in A\}I\{Y\in B\}=P(X\in A,Y\in B), \end{align*} so that $P(X\in A,Y\in B)=P(X\in A)P(Y\in B)$.

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  • 1
    $\begingroup$ Iosif, if $f(x,y) = I\{x \in A\} I\{y \in B\}$, then $\varphi(y)$ need not be equal $P(X \in A) I\{y \in B\}$! You are assuming independence here... $\endgroup$ – Mateusz Kwaśnicki Apr 13 at 11:25
  • $\begingroup$ What happens $P_X$ and $P_Y$ supported on finite (or compact) sets and let $f(x,y)=1$? $\endgroup$ – DCM Apr 13 at 12:24
  • $\begingroup$ I read this question as being about the functional $\mu_{X,Y,f}:\psi\longmapsto \int\int\psi(y)f(x,y)(dP^{X,Y}(x,y) - dP^X(x)dP^Y(y))$ for a fixed choice of $f$. $\endgroup$ – DCM Apr 13 at 12:25
  • $\begingroup$ I was going to use the word distribution instead of `functional' there for a second... $\endgroup$ – DCM Apr 13 at 12:28
  • $\begingroup$ @MateuszKwaśnicki : Mateusz, I don't know why you said "need not". I was certainly not assuming any independence at that point. If $f(x,y)=I\{x\in A\}I\{y\in B\}$ for all real $x,y$, then for each $y\in\mathbb R$ we have $\varphi(y)=Ef(X,y)=EI\{X\in A\}I\{y\in B\}=P(X\in A)I\{y\in B\}$. Here, $y$ is just a real number (not a random variable), and hence $I\{y\in B\}$ is also just a real number. I have added this little detail to the answer. Please let me know whether this convinces you. $\endgroup$ – Iosif Pinelis Apr 14 at 1:26
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This isn't an answer, just a (hopefully correct and useful) reformulation of the question

Let $f$ be as stipulated (and assumed fixed throughout). The equation $E(f(X,Y)|Y) = u(Y)$ is just a way of saying that $u$ satisfies

$$ \int f(x,y)\psi(y)dP^{X,Y}(x,y) = \int u(y)\psi(y)dP^Y(y) $$

for all $\psi$ in a suitably large class of test functions. The question here is under what circumstances is the function $u$ defined by

$$ u(y) =\int f(x,y)dP^X(x) $$

a solution to all these equations. That is, when is it true that

$$ \int\int f(x,y)\psi(y)dP^{X,Y}(x,y) = \int \left(\int f(x,y)dP^X(x)\right)\psi(y)dP^Y(y) $$

for all test functions $\psi$? Re-arranging this equation slightly, our question becomes: when is it true that

$$ \int \int \psi(y)f(x,y)\left( dP^{X,Y}(x,y) - dP^X(x)dP^Y(y)\right) = 0 $$

for all test functions $\psi$?.

Edit: I don't disagree with anything in Iosif's answer, but my initial feeling (before you accepted his answer) was that you were asking a slightly different question. Whether anything interesting can be said for my question is perhaps a matter for another occasion.

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