Conditional Expectation of X conditioned on sigma algebra generated by X and Y where X,Y are independent

Question

I am sure I understood this before, but forgot how I made sense of it, could someone please let me know why is the following correct?

$$E[X\mid \sigma(X+Y,X)] = E[X\mid\sigma(X,Y)] = E[X\mid\sigma(X)] = X$$

where $X,Y$ are two independent random variables.

Thanks in advance.

Answer 1

You can jump directly from $E[ X\mid \sigma(X+Y,X)]$ to the answer $X$, because of the following:

If $X$ is $\cal F$ measurable, then $E[X\mid {\cal F}] = X$.

Proof: This follows from the definition of conditional expectation -- a thing $U$ qualifies as the conditional expectation $E[X\mid {\cal F }]$ if $U$ is $\cal F$-measurable and if $E[U I(F)]=E[X I(F)]$ for all $F\in\cal F$. Clearly $X$ qualifies, given the assumption that it is $\cal F$ measurable.

So in your example it is clear that $X$ is $\sigma(X,X+Y)$ measurable, since $X$ is right there in $\sigma(X,X+Y)$. Independence is not required, btw.