Conditional expectation for nonnegative variables

Assume AC. For measurable $X:\mathrm{\Omega }\to [0,\mathrm{\infty }]$, select versions $U_n$ of $E[X\wedge n\mid \mathcal{G}]$, $n\ge 1$. Outside one $\mathcal{G}$-measurable null set they are nonnegative and increasing. Set all of them to zero on that set. Define $E[X\mid \mathcal{G}]$ to be the almost-sure class of ${\lim }_n{U}_n$, allowing $+\mathrm{\infty }$.

The integrable classes come from Conditional expectation as an ae class. Their order is Basic algebra and order properties of conditional expectation. The union of the measurable sets where a nonnegativity or consecutive-order condition fails is a measurable null set. The limit is measurable by Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable. The The Axiom of Choice supplies both inherited RN choices and the countable selection of versions. The event-integral characterization and independence of truncations are the well-definedness obligations recorded in justified_by.

Van der Vaart Lemma 1.10(i), printed p.4; Durrett Theorem 4.1.9(c), printed pp.210–211, supplies the integrable case. The local next theorem proves the extended-valued definition.