Convergence of proposed approximations to conditional expectation

Question

Let $X,Y$ be real random variables on $(\Omega, \mathscr{F}, P)$ with $E[|X|] < \infty$. Let $Z_1, Z_2, \dots$ be a sequence of proposed approximations of $E[X|Y]$ defined by

$$Z_n(\omega) = \sum_{k=-\infty}^ {k=\infty} \frac{E[X I\{\frac{k}{2^n} \leq Y \lt \frac{k+1}{2^n}\}] I\{\frac{k}{2^n} \leq Y \lt \frac{k+1}{2^n}\}}{P(\frac{k}{2^n} \leq Y \lt \frac{k+1}{2^n})}$$

                   =some arbitrary number if the denominator is zero.

I have to show that $Z_n \to E[X| Y]$ a.s.

Soln. Approach:

I tried to use the fact for any $A\in \sigma(Y)$ if $\int_A Z_n \to \int_A X $ then $Z_n \to E[X| Y]$ a.s. But there is some problem to change limit and integrals.

Answer 1

We can use the following approach:

• we check that $$\tag{*}\int_A Z_n\to \int_AX$$ when $A:=Y^{-1}(I)$, where $I$ is an interval of the form $(m2^{-N}, m'2^{-N})$.
• We extend this to the set $A:=Y^{-1}(I)$ where $I$ is an arbitrary interval.
• Then we consider the case $A:=Y^{-1}(O)$ where $O$ is an open set.
• The general case.