# Conditional expectation for independent random variables

I have a question about the conditional expectation with some independence conditions for random variables and $$\sigma$$-fields.

For a random variable $$X$$ with $$E|X| < \infty$$, if $$Y_1$$ and $$Y_2$$ are random variables such that $$\sigma(X,Y_1)$$ and $$\sigma (Y_2)$$ independent, then I want to prove the following. $$E\left(X | Y_{1}, Y_{2}\right)=E\left(X | Y_{1}\right) \quad \text { a.s. }$$

It seems very intuitive since $$Y_2$$ information is useless for $$X$$. But I don't know how to prove it. Can anyone help me?

Since RHS is measurable w.r.t $$\sigma (Y_1,Y_2)$$ we only have to show that $$E I_EX=EI_E E(X|Y_1)$$ for every set $$E \in \sigma (Y_1,Y_2)$$. By a standard argument using the $$\pi -\lambda$$ theorem we can reduce this to the case when $$E$$ has the form $$Y_1^{-1} (A) \cap Y_2^{-1}(B)$$ where $$A$$ and $$B$$ are Borel sets in $$\mathbb R$$. Now $$EXI_{Y_1^{-1}(A)}I_{Y_2^{-1}(B)}=EXI_{Y_1^{-1}(A)} EI_{Y_2^{-1}(B)}$$ by the independence assumption. Also $$E E(X|Y_1)I_{Y_1^{-1}(A)} I_{Y_2^{-1}(B)}$$ $$=E(E(XI_{Y_1^{-1}(A)}|Y_1) I_{Y_2^{-1}(B)})$$ $$=E(E(XI_{Y_1^{-1}(A)}|Y_1)) EI_{Y_2^{-1}(B)}$$ $$=EXI_{Y_1^{-1}(A)} EI_{Y_2^{-1}(B)}.$$ This completes the proof.