I'm studying stochastic and I'm stuck at the following problem: I ask myself how to compute this conditional expectation:
Let $X$, $Y$ be two independent random variables in $L^1$. What is $\mathbb{E}[X | \sigma(X+Y)]$?
For me it's clear that $\mathbb{E}[X | \sigma(X)] = X$, because $X$ is $\sigma(X)$-measurable and $\mathbb{E}[X | \sigma(Y)] = \mathbb{E}[X]$, because $X$ and $Y$ are independent and therefore $X$ is independent of $\sigma(Y)$ .
Could someone give me a hint?