= Conditional covariance
{title2=$\operatorname{Cov}(X,Z\mid\mathcal G)$}
For square-integrable random variables, $\operatorname{Cov}(X,Z\mid\mathcal G)=\mathbb E[XZ\mid\mathcal G]-\mathbb E[X\mid\mathcal G]\mathbb E[Z\mid\mathcal G]$. 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>.
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