Conditional probability given a sigma algebra

Assume AC for conditional-expectation existence. On a probability space $(\mathrm{\Omega },\mathcal{F},P)$, for a sub-sigma-algebra $\mathcal{G}\subseteq \mathcal{F}$ and $A\in \mathcal{F}$, define $P(A\mid \mathcal{G})=E[1_A\mid \mathcal{G}]$ as the almost-everywhere class in Conditional expectation as an ae class. The indicator is integrable since $E\mathrm{\mid }{1}_A\mathrm{\mid }=P(A)\le 1$, so Conditional expectation exists by radon nikodym applies. A version is a real integrable $\mathcal{G}$-measurable function whose integral over each $H\in \mathcal{G}$ is $P(A\cap H)$.

The The Axiom of Choice hypothesis is inherited from the RN existence proof's maximizing-sequence and Hahn-decomposition selections. It does not select a canonical representative. In particular, choosing a version separately for each event A has not constructed a probability measure on the event space at any fixed sample point. The constant zero and one functions satisfy the testing identities for the empty and whole events respectively. No completion of $\mathcal{G}$ is part of this definition.