It's ok for the definition of .
For the definition of a real valued random variable, it's correct for the first part : it's a function from to . But I don't agree with the second part : for , there is a unique . It's just a function.
If you want to go further, you'll have to talk about measurability of the random variable. I don't know if you know this though.
No, in general, A will be another random variable, not an event. At least that's how conditional expectation works.I am trying to understand two concepts
1. E(X) - Expected value of X
2. E(X|A) - Conditional expectaion under event A
I think in 2 above, we have not changed the way X was defined on the sample space, only thing that changed is probabiltiy measure - Is this understanding correct?
E(X|A) will be another random variable. You can keep the probability measure, because the sample space isn't much modified.
*In a non-formal way* The thing that changes is that in the "information" of X, we only keep the parts that are "related" to A.
E(X|A) is viewed as the orthogonal projection of X in
I guess this is a bit abstract, but since I don't know what your background is, I just hope you will find your answer among all this stuff