Suppose we have three random variables $Y$, $X$, and $Z$, each of which is univariate. Define an indicator function $1(|X-x|\leq a)=1$ if $x-a\leq X\leq x+a$ and $0$ otherwise, for some positive values $a$ and $x$.
Question: What is conditional expectation of $E(Y|1(|X-x|\leq a)Z)?$ That is, the expectation of $Y$ given the condition that a subset of Z for which $X-x$ is around $a$ is used.
My guess is that, since the indicator defines a subset for which the condition is defined in the expectation, we might have
$$ E(Y|1(|X-x|\leq a)Z)=\int_y y\frac{f(y,1(|X-x|\leq a)Z)}{f(1(|X-x|\leq a)Z)}dy. $$
What prevents me from going further is the density that involves the indicator function, i.e., $f(y,1(|X-x|\leq a)Z)$ and $f(1(|X-x|\leq a)Z)$. I guess these densities should be expressed as probabilities because the condition is actually an event $A=\{Z: Z\neq 0 \ \text{for} \ |X-x|\leq a \}$. Can anyone provide an explicit expression on the conditional expectation above?
Thanks.