0 Prerequisites
0.1 Lebesgue conditional expectation
Let \((X, \mathcal X)\) be a measurable space, let \(\mathcal B\) be a sub \(\sigma \)-algebra of \(\mathcal X\).
The conditional expectation of a \(\mathcal X\)-measurable function \(f : X \to [0, \infty ]\) is
\[ \mu [f | \mathcal B] = ?? \]
If \(f : X \to [0, \infty ]\) is a \(\mathcal X\)-measurable function, then \(\mu [f | \mathcal B]\) is the \(\mu \)-ae unique \(\mathcal B\)-measurable function \(X \to [0, \infty ]\) such that
\[ \int _B \mu [f | \mathcal B]\ \partial \mu = \int _B f\ \partial \mu \]
for all \(B \in \mathcal B\).
Proof
Standard machine.