Can $X$ be treated as a constant on the left hand side of the conditioning bar in the expression $P\left(\cdot\mid X\right)$?

Question

Consider the following example. Let $\left(\Omega,\mathcal{F},P\right)$ be a probability space and let $X$ and $Y$ be random variables defined in this space. Suppose $X$ and $Y$ are independent and $Y\sim\mathrm{Exp}\left(1\right)$. It can be shown that $P\left(Y>X\mid X\right)=e^{-X}$ $P$-a.s., essentially treating $X$ as a constant on the left-hand side of the conditioning bar.

Can $X$ always be treated as a constant on the left hand side of the conditioning bar in the expression $P\left(\cdot\mid X\right)$? In other words, is the following statement true?

Let $\left(\Omega,\mathcal{F},P\right)$ be a probability space and let $X$ and $Y$ be random variables defined in this space.

Denote by $\mathfrak{B}$, $\mathfrak{B}_2$ the Borel fields on the line/plane, respectively. Suppose $K:\Omega\times\mathfrak{B}\rightarrow\left[0,1\right]$, $Q:\Omega\times\mathfrak{B}_2\rightarrow\left[0,1\right]$ are two stochastic kernels that are versions of the conditional distributions $P\left(Y\in\cdot\mid X\right)$, $P\left(\left(X,Y\right)\in\cdot\mid X\right)$, respectively.

Let $A\in\mathfrak{B}_2$, and for each $r\in\mathbb{R}$, denote by $A_r$ the $r$-section of $A$ keeping the first coordinate fixed, i.e. $A_r := \left\{\left(a,b\right)\in A\mid: a = r\right\}$.

Is it true that

$$ Q\left(\omega,A\right)=K\left(\omega,A_{X\left(\omega\right)}\right) $$

$P$-a.s.?

Answer 1

Yes, the statement is true. It can be shown by considering the collection $S\subseteq\mathfrak{B}_2$ of all sets $A\in\mathfrak{B}_2$ for which

1. the function $\omega\in\Omega\mapsto K\left(\omega, A_{X\left(\omega\right)}\right)$ is $\mathcal{F}/\mathfrak{B}$-measurable.
2. $Q\left(\omega,A\right)=K\left(\omega,A_{X\left(\omega\right)}\right)$ $P$-a.s.

and showing that $S = \mathfrak{B}_2$ using Dynkin's $\pi$-$\lambda$ lemma, by showing that

1. $S$ contains all the open intervals of the form $\left(a,b\right)\times\left(c,d\right)$ with $a,b,c,d\in\left[-\infty,\infty\right]$ (hence, in particular, $\mathbb{R}^2\in S$).
2. Whenever $A\in S$, $A^c\in S$ (with $A^c:=\mathbb{R}^2\setminus A$).
3. Whenever $\left(A_n\right)_{n=1}^\infty$ is a sequence of pairwise disjoint sets each of which belongs to $S$, then $\bigcup_{n=1}^\infty A_n\in S$.

These three steps prove that $S$ is a $\sigma$-algebra that contains the open rectangles in $\mathbb{R}^2$, and so $\mathfrak{B}_2\subseteq S$.