Conditional monotone convergence theorem ($Y_n\uparrow Y⟹\mathbb{E}[Y_n\mid \mathcal{G}]\uparrow \mathbb{E}[Y\mid \mathcal{G}]$)

For nonnegative random variables $Y_n\uparrow Y$, the corresponding nonnegative conditional expectations increase almost surely to $\mathbb{E}[Y\mid \mathcal{G}]$, allowing the extended value $+\infty$. When $Y$ is an integrable random variable, all conditional expectations can be taken in their ordinary finite sense.

To prove this integrable case, choose versions simultaneously ordered outside a single null set, and put $V={\lim }_n\mathbb{E}[Y_n\mid \mathcal{G}]$. This limit is $\mathcal{G}$-measurable. For every $A\in \mathcal{G}$, the ordinary monotone convergence theorem twice givesIn particular $\mathbb{E}V=\mathbb{E}Y<\infty$, and this is exactly the defining property of conditional expectation. For a general nonnegative $Y$, define its extended conditional expectation by increasing bounded truncations; the same nonnegative integral argument proves the statement.