The most general definition of the conditional expectation that I know is the following:
Let $(\Omega, \mathcal{F}, P )$ a measure space and $X: \Omega \to \mathbb{R}$ a random variable. Given a sigma algebra $\mathcal{G}\subset \mathcal{F} $. The conditional expectation $E(X \mid \mathcal{G})$ is an unique random variable that:
- $E(X \mid \mathcal{G})$ is $\mathcal{G}-$measurable
- $\int_B E(X \mid \mathcal{G}) dP = \int_B X dP, \quad \forall B \in \mathcal{G}$.
When $ \mathcal{G}$ is the $\sigma-$algebra generated by a random variable $Y:\Omega \to \mathbb{R}$, we adopt the notation: $E(X \mid Y)$.
I would like to generalize this definition for random vectors $X = (X_1,...,X_n)$.
The unconditional vector is straightforward: if $\mu_i = E(X_i)$, then $E(X)= (\mu_1,..., \mu_n)$.
In the multivariate case, for example, if $Y = (Y_1,...,Y_n)$, can I conclude the following? \begin{equation} E(X\mid Y) = (E(X_1 \mid Y_1), \ldots, E(X_n \mid Y_n)) \end{equation}
How about the case when $Y = (Y_1,...,Y_m)$ with $m>n$?