Conditional Expectation

Although the CEF is random variable, it's not "defined" pointwise.

\[\begin{align*} &\mathbb{E}[Y|X] :: \Omega \to \mathcal{R} \\ & \mathcal{I} \ \mathbb{P} \ A \ Y = \mathcal{I} \ \mathbb{P} \ A \ \mathbb{E}[Y|X] \quad \forall \ A \in \sigma(X) \end{align*}\]

The law of iteration of expectations is part of the definition

\[\begin{align*} & \mathcal{I} \ \mathbb{P} \ \Omega \ Y = \mathcal{I} \ \mathbb{P} \ \Omega \ \mathbb{E}[Y|X] \end{align*}\]