Conditional covariance ($\mathrm{Cov}(X,Z\mid \mathcal{G})$)

For square-integrable random variables, $\mathrm{Cov}(X,Z\mid \mathcal{G})=\mathbb{E}[XZ\mid \mathcal{G}]-\mathbb{E}[X\mid \mathcal{G}]\mathbb{E}[Z\mid \mathcal{G}]$. It is a $\mathcal{G}$-measurable random variable. Replacing $X$ by its conditional expectation given a larger sigma-field leaves this covariance unchanged when $Z$ is measurable with respect to that larger sigma-field. This observation underlies recovery of a martingale-transform integrand by conditional covariance.