I have three independent uniforms variables, $U, V, W$ that take their values in $[0,1]$. I also have $X = \max(U, W)$ and $Y = \max(V,W)$.
I need to compute the conditional expectation of $W$ knowing (the $\sigma$-algebra generated by) $X$ and $Y$. I have proven that it can be written as $f(X,Y)$ with $f$ measurable.
Is there any way to compute this without too much calculations? I have already computed $E(W \mid X)$ but I can't find any simple way to use this. Do I have to make the full calculation?