= Conditional monotone convergence theorem
{title2=$Y_n\uparrow Y\Longrightarrow\mathbb E[Y_n\mid\mathcal G]\uparrow\mathbb E[Y\mid\mathcal G]$}
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 gives
$$
\mathbb E[\mathbf1_AV]
=\lim_n\mathbb E[\mathbf1_A\mathbb E[Y_n\mid\mathcal G]]
=\lim_n\mathbb E[\mathbf1_AY_n]
=\mathbb E[\mathbf1_AY].
$$
In particular $\mathbb EV=\mathbb EY<\infty$, 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.
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