= Conditional expectation preserving a distribution
{title2=$\mathcal L(\mathbb E[X\mid\mathcal G])=\mathcal L(X)\Longrightarrow X=\mathbb E[X\mid\mathcal G]\text{ a.s.}$}
For a real <integrable random variable> $X$, a <conditional expectation> $Y=\mathbb E[X\mid\mathcal G]$ with the same <probability distribution> as $X$ must equal $X$ <almost surely>. No second moment is needed. For any real $c$, the nonnegative <random variable> $D_c=|X-c|-\operatorname{sign}(Y-c)(X-c)$ has <expected value> zero, by conditioning and equality of the two absolute moments. Thus $Y>c$ implies $X\geq c$, $Y<c$ implies $X\leq c$, and $Y=c$ implies $X=c$, outside a null set. Using a <countable> family of rational $c$ separates any unequal real values and proves the claim. Equality of the probabilities of $X=c$ and $Y=c$ also gives equality of their zero sets and hence equality of the two sign functions for every fixed $c$.
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