Conditional Expectation

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

$$\begin{array}{rl}& \mathbb{E}[Y|X]::\mathrm{\Omega }\to \mathcal{R}\\ & \mathcal{I}\mathbb{P}AY=\mathcal{I}\mathbb{P}A\mathbb{E}[Y|X]\mathrm{\forall }A\in \sigma (X)\end{array}$$

The law of iteration of expectations is part of the definition

$$\begin{array}{rl}& \mathcal{I}\mathbb{P}\mathrm{\Omega }Y=\mathcal{I}\mathbb{P}\mathrm{\Omega }\mathbb{E}[Y|X]\end{array}$$