What is the value of this expectation?

Question

I have a conditional expectation question: how to calculate this expectation?

$$E[X-E(X|Y)|Y]$$

Answer 1

Conditional expectations are linear: $$E[X-E(X|Y)|Y]=E[X|Y]-E[E(X|Y)|Y]$$ By the tower property (requiring that the sigma algebra of the outer expectation is in the sigma algebra of the inner expectation). $$E[X|Y]-E[E(X|Y)|Y]=E[X|Y]-E[X|Y]=0$$

Answer 2

Hint:

$$E\bigg(E(X|Y)|Y\bigg)=E(g(Y)|Y)=g(Y)=E(X|Y)$$