Expressing the probability of a random vector being in a random set as a conditional expectation

Question

Let $X$ be a random vector in $\mathbb{R}^n$ and let $\mathbb{P}_X$ be its distribution. Let $\{A_y : y \in \mathbb{R}^m\}$ be some collection of Borel subsets of $\mathbb{R}^n$ and let $Y$ be a random vector in $\mathbb{R}^m$. Is it true that $\mathbb{P}_X(A_Y) = \mathbb{E}[\mathbb{I}_{A_Y}(X) \mid Y]$, where $\mathbb{I}_{A_y}$ is the indicator function of $A_y$? It seems reasonable to me, but I don't see how to prove this.

Answer 1

If $X$ and $Y$ are independent and assuming a suitable measurability condition on the map $y\mapsto A_y$, the answer is yes.

To begin with, we formulate the measurability of $y\mapsto A_y$ in terms of the map $(x, y) \mapsto \mathbb{I}[x \in A_y]$. That is, we say that $y \mapsto A_y$ is measurable iff the map $(x, y) \mapsto \mathbb{I}[x \in A_y]$ is jointly measurable.

Now, we assume

1. $X$ and $Y$ are independent, and
2. $y \mapsto A_y$ is measurable.

Then

\begin{align*} \mathbb{E}[\mathbb{I}_{A_Y}(X) \mid Y] &= \mathbb{E}[\mathbb{I}[X \in A_Y] \mid Y] \tag{1} \\ &= \mathbb{E}[\mathbb{I}[X \in A_y]]_{y = Y} \tag{2} \\ &= \mathbb{P}_X(A_y)_{y = Y} \tag{3}\\ &= \mathbb{P}_X(A_Y) \tag{4} \end{align*}

• $\text{(1)}$ : The indicator function is rephrased.
• $\text{(2)}$ : This holds because of the independence of $X$ and $Y$.
• $\text{(3)}$ : Change of variables formula.
• $\text{(4)}$ : Substituting $y = Y$.