Conditional probability, $E[E(X\mid Y)\mid Y,Z]=?$

Question

I know that $E[E(X\mid Y,Z)\mid Y]=E(X\mid Y)$.

What if $E[E(X\mid Y)\mid Y,Z]=?$ It has any meaning?

Answer 1

$E(X|Y)$ is $Y$-measurable and thus $Y,Z$-measureable so can still be pulled out of the conditional expectation. So $E(E(X|Y)|Y,Z) = E(X|Y)$

It's like asking about $E(f(Y)|Y,Z)$. The $Z$ might seem to complicate things, but the fact that we're conditioning on $Y$ means we can treat the $f(Y)$ as constant and pull it out of the expectation, giving $f(Y)E(1|Y,Z) =f(Y).$