 Equality case for conditional second moments

If $Y$ is a real square-integrable random variable and $X=\mathbb{E}[Y\mid \mathcal{G}]$, then $\mathbb{E}[XY]=\mathbb{E}{X}^2$ and the displayed identity follows by expansion. Thus equality of the second moments forces $X=Y$ as an almost sure equality. This is the equality case of the conditional Jensen inequality for the strictly convex function $z\mapsto z^2$.