For all $A \in \sigma\{X+Y\}$, we have $\mathbb{E}[X:A] = \mathbb{E}[Y:A]$?

Question

Let $X$ and $Y$ be independent identically distributed random variables. How can one show that for all $A \in \sigma\{X+Y\}$, we have $\mathbb{E}[X:A] = \mathbb{E}[Y:A]$?

Answer 1

Any $A \in \sigma (X+Y)$ is of the form $A=(X+Y)^{-1} (E)$ for some Borel set $E$ in $\mathbb R$. Now write the integrals as integrals over the product space and apply the transformation $(x,y) \to (y,x)$. Since $X,Y$ is i.i.d. this transformation does not change the product measure.