I am trying to understand the proofs of the properties of conditional expectation.
I first start with the definition of conditional expectation:
let $X$ be an integrable r.v. on the probability space $(\Omega,\mathcal F,\mathbb P)$ and $\mathcal G\subset \mathcal F$ a sigma-algebra.
Then a r.v. $Y=\mathbb E(X|\mathcal G)$, $\mathcal G$-measurable function for which holds $\mathbb E(XI_A)=\mathbb E(YI_A)$ for each $A\in \mathcal G$ is called conditional expectation of X given $\mathcal G$.
Now, if I want to prove the "pull out what is known" I have to prove: $\mathbb E(XY|\mathcal G)=Y\mathbb E(X|\mathcal G)$ if Y is $\mathcal G$-measurable
How to prove this? Do I have to show that $\mathbb E(XY|\mathcal G)$ is $\mathcal G$-measurable and that $\mathbb E(XYI_A)=\mathbb E(\mathbb E(XY|\mathcal G)I_A) $?
I have no clue, I see everywhere on the proofs I find that "clearly $ Y\mathbb E(X|\mathcal G)$ is $\mathcal G$-measurable", why?
If I use the definition of conditional expectation I may say that $\mathbb E(X|\mathcal G)$ is $\mathcal G$-measurable, and that Y is same by the assumptions, but I don't know what happens to their product.
The second point I don't understand is that we can prove the equality if we show $\mathbb E(Y\mathbb E(X|\mathcal G)I_A)=\mathbb E(XYI_A)$. The steps after this I can understand but I don't know why I have to show this.
To me these two quantities can be re-written as $\mathbb E (\mathbb E(XY|\mathcal G)I_A)=\mathbb E(YXI_A)$
I tried to ask the prof and he said that we need to show that the property is a conditional expectation by checking the measurability and then the equation on top for each A, but for me it's very confusing.
Best Answer
You'll want to do it in four parts: prove it for constant functions, simple functions, positive functions, then all functions. I'm pretty sure you'll also need the dominated convergence theorem. Many examples of this 4-part style proof can surely be found in whatever textbook you're using (if you're not using one, let me know and I'll link you one).
This should've been a comment but I didn't have enough rep to make one.. sorry about that. Hopefully it helps.