Expressing $E[h(X,Y)|X]$ as an integral over $Y$

Question

Let $X$ and $Y$ be two random variables taking values in $(E,\mathcal{E})$ and $(F,\mathcal{F})$, respectively, such that the conditional law of $Y$ given $X$ exists, denoted $\mathrm{P}_{Y|X}$. That is to say, for all $g\colon F \to \mathbb{R}_+$ measurable,

$$\mathrm{E}[g(Y)|X] = \int_{F} g(y) \mathrm{P}_{Y|X}(\mathrm{d}y)~.$$

Now, is it always true that if $h\colon E \times F \to \mathbb{R}_+$ is measurable, then $$\mathrm{E}[h(X,Y)|X] = \int_{F} h(X,y) \mathrm{P}_{Y|X}(\mathrm{d}y)~?$$

I know that it is the case for variables with densities (using Fubini's theorem, $\mathrm{P}_{Y|X}$ can actually be deduced from the joint density), or if $X$ is discrete (i.e. $E$ is countable). What about the general case?

Answer 1

Actually one also have to check that the function we defined is $\sigma(X)$-measurable. Let then $$H = \Big\{h\colon E\times F \to \mathbb{R}_+ \Big| \: x \mapsto \int_F h(x,y) \mathrm{P}_{Y|X=x}(\mathrm{d}y) \text{ is $\mathcal{E}$-measurable}, \\ \text{ and } \mathrm{E}[h(X,Y)|X] = \int_F h(X,y) \mathrm{P}_{Y|X}(\mathrm{d}y) \Big\}~.$$

Let us first show that $H$ contains all simple functions. To this end, let $\mathcal{C} = \{ C \in \mathcal{E} \otimes \mathcal{F} | 1_{C} \in H \}$. We have:

• $\mathcal{C}$ contains the rectangles. Indeed, if $(A,B) \in \mathcal{E} \times \mathcal{F}$, $1_{A \times B}\colon (x,y) \mapsto 1_A(x) 1_B(y)$, and then $x \mapsto 1_A(x) \mathrm{P}_{Y|X=x}(B)$ is measurable (by definition of the conditional law and measurability of $A$), and finally $\mathrm{E}[1_A(X) 1_B(Y)|X] = 1_A(X) \mathrm{E}[1_B(Y)|X]$, which leads to $1_A(X) \int_F 1_B(y) \mathrm{P}_{Y|X}(\mathrm{d}y) = \int_F 1_{A \times B}(X,y) \mathrm{P}_{Y|X}(\mathrm{d}y)$.
• Now if $C, C' \in \mathcal{C}$, $C \subset C'$, then $1_{C'\setminus C} = 1_{C'} - 1_C$, and by linearity we deduce that $C'\setminus C \in \mathcal{C}$.
• Finally, if for all $n \in \mathbb{N}$, $A_n \in \mathcal{C}$ and $A_n \subset A_{n+1}$, then $1_{\cup_n A_n} = \lim_N \uparrow \sum_{n = 0}^N 1_{A_{n+1}\setminus A_n}$, and we conclude by measurability of limit, and monotone convergence of conditional expectation, that $\cup_n A_n \in \mathcal{C}$.

Altogether, $\mathcal{C}$ is a Dynkin system containing the rectangles, and by the $\pi$-$\lambda$ theorem, $\mathcal{C} = \mathcal{E} \otimes \mathcal{F}$.

It is now very easy to conclude that $H$ contains all positive measurable functions on $E \times F$, using the exact same third argument above.