Proof Linearity of Conditional expectation

Question

How we can proof that:
$E[X - Y|W] = E[X|W]-E[Y|W]$
I try to use the definition of Conditional Expectation:
$E[X|Y=y]= \sum_x \cdot p(x|y)$ and then to substitute $X=A-B$
Is it the right way? And how we continue from here?

Answer 1

You did not explicitate if your rv's are contintuous or not. Thus I assume they are continuous:

$$E[X-Y|W]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}(x-y)f_{XY|W}(x,y|w)dxdy=$$

$$=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}xf_{XY|W}(x,y|w)dxdy-\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}yf_{XY|W}(x,y|w)dxdy=$$

$$=\int_{-\infty}^{+\infty}xf_{X|W}(x|w)dx-\int_{-\infty}^{+\infty}yf_{Y|W}(y|w)dy=E(X|W)-E(Y|W)$$