Conditional expectation implied by expectation of product

Question

For uni I have got the following assignment, and I've gotten stuck at (b). I have proven that E[1_{bi}X] = E[1_{bi}X], but don't know how I would get from this to the required conclusion. Does anyone have hints on how to solve this? Any help is appreciated.

Thanks in advance!!

Answer 1

The definition of $Y=E[X \mid \mathcal{G}]$ is that $Y$ is $\mathcal{G}$-measurable and satisfies $$E[1_A X] = E[1_A Y], \; \forall A \in \mathcal{G}.$$

You've already showed that this holds when $A$ is one of the $B_i$, but you need to show it holds for other sets in $\mathcal{G}$. Kavi Rama Murthy is pointing out that the other sets of $\mathcal{G}$ are unions of the $B_i$. (Can you justify why?) Then, for example, can you check that the above equality holds when $A$ is $B_1 \cup B_2$?