Consider for the trvial $\sigma$ - field $\mathcal{F}_0 = \{\emptyset , \Omega\}$, What is Conditional expectation of the following in the following cases when $A = \emptyset$ and $A = \Omega$ ???
? Can someone please help me fill in the ?? below, as this would help improve my understanding a lot ?
$\int_? E[X | \mathcal{F}_0]1_A dP = ? \;\; \forall A \in \mathcal{F}_0$
Question 2: And What if I just condition on the $\sigma$-field,
$ \int_? E[X | \mathcal{F}] dP = ? $
For the second question, I guess it is = X right? Since X is already $\mathcal{F}$- measurable by definition of random variable, if given the $\sigma$ - feld $\mathcal{F}$, everything is known, there is no randomness in X.