= L1 contraction of conditional expectation
{c}
{title2=$\|\mathbb E[Y\mid\mathcal G]\|_1\leq\|Y\|_1$}
For an <integrable random variable> $Y$, the <conditional expectation> satisfies $|\mathbb E[Y\mid\mathcal G]|\leq\mathbb E[|Y|\mid\mathcal G]$ <almost surely>. Taking <expected values> gives the <L1 contraction of conditional expectation>. Applying it to $Y_n-Y$ shows that <convergence in L1> is preserved by <conditional expectation> with respect to any fixed <sigma-algebra>.
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