Conditional expectation with 3 random variables

Question

If we know $E(Y) = E( E( Y \mid X ) )$, and $X$ and $Y$ are independent, I am wondering whether $E(Y)$ is still equal to $E( E( Y \mid X, f(X) ) )$, where $f(X)$ is an approximation of $Y$?

Answer 1

Hint:

For any measurable function $f$

$$E(Y|X, f(X))=E(Y|X)$$(it does not require $X$ and $Y$ be independent)

It is enough to show for any measurable function $f$,

$$\sigma(X, f(X))=\sigma(X)$$.

Answer 2

Yes. If you are familiar with conditional expectaions given a sigma field then $EY=E(Y|\sigma (X))$ which implies that $EY=E(Y|X,f(X))$ for any measurable function $f$.