Suppose $(\Omega, \mathcal{F}, P)$ is a probability space and $(U, \mathcal{\Sigma})$ is a measurable space. There seem to be two ways of defining the conditional expectation of a r.v. $X: \Omega \rightarrow \mathbb{R}$ given another r.v. $Y: \Omega \rightarrow U$, denoted as $E(X\vert Y)$:
- As a $\sigma(Y)$-measurable mapping from $\Omega$ to $\mathbb{R}$, defined as: $E(X\vert Y) = E(X \vert \sigma(Y)). $ where $\sigma(Y)$ is the sigma algebra of r.v. $Y$, which I think is also denoted as $Y^{-1}(\mathcal{\Sigma})$?
As a $\mathcal{\Sigma}$-measurable mapping from $U$ to $\mathbb{R}$, defined as follows (from Wikipedia):
Define measure Q on U to be the probability measure induced by $Y$ on $(U, \mathcal{\Sigma})$, as $Q(B) = P(Y^{−1}(B)), \forall B \in \mathcal{\Sigma}$.
Define $E(X \vert Y)$ to be the integrable function $g:U \rightarrow \mathbb{R}$ such that
$ \int_{Y^{-1}(B)} X(\omega) \ d \operatorname{P} = \int_{B} g(u) \ d \operatorname{Q}, \forall B \in \mathcal{\Sigma}.$
If I am correct, this definition is related to the first one as: $E(X \mid Y) \circ Y= E\left(X \mid Y^{-1} \left(\Sigma\right)\right). $
I was wondering which of the above two is the definition of $E(X \mid Y)$?
Thanks and regards! References (links or books) will also be appreciated!