 Conditional expectation preserving a distribution

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 $D_c=|X-c|-\mathrm{sign}(Y-c)(X-c)$ has expected value zero, by conditioning and equality of the two absolute moments. Thus $Y>c$ implies $X\ge c$, $Y<c$ implies $X\le c$, 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 $c$.