= Uniform integrability of conditional expectations
For one <integrable random variable> $Y$, the family $Z_{\mathcal G}=\mathbb E[Y\mid\mathcal G]$, over arbitrary <sigma-algebras> $\mathcal G$, has <uniform integrability>. On $A=\{|Z_{\mathcal G}|>K\}$, the defining property of <conditional expectation> gives $\mathbb E[|Z_{\mathcal G}|\mathbf1_A]\leq\mathbb E[|Y|\mathbf1_A]$, while $\mathbb P(A)\leq\mathbb E|Y|/K$ by the <Markov inequality>. The <integrable random variable> $Y$ has uniformly small absolute integrals over <events> of sufficiently small probability.
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