 Bayes formula for conditional expectation

Suppose $\tilde{\mathbb{P}}$ is a probability measure with $d\tilde{\mathbb{P}}=Zd\mathbb{P}$, so $Z\ge 0$ and $\mathbb{E}Z=1$. For a sub-sigma-algebra $\mathcal{G}$, write $Z_{\mathcal{G}}=\mathbb{E}[Z\mid \mathcal{G}]$. If $X$ is integrable under $\tilde{\mathbb{P}}$, then on $\{Z_{\mathcal{G}}>0\}$,This is Bayes formula for conditional expectation. It changes the underlying probability measure as well as conditioning on the available information. The set where $Z_{\mathcal{G}}=0$ has zero $\tilde{\mathbb{P}}$-probability, so a version there can be chosen arbitrarily.

To verify the formula, let $Y$ denote the ratio. For $A\in \mathcal{G}$, measurability of $Y$ and the tower property give ${\int }_{A}Yd\tilde{\mathbb{P}}=\mathbb{E}[1_{A}ZY]=\mathbb{E}[1_A{Z}_{\mathcal{G}}Y]=\mathbb{E}[1_{A}ZX]={\int }_{A}Xd\tilde{\mathbb{P}}$. These are exactly the defining conditional-expectation identities. For a filtration, choosing $\mathcal{G}={\mathcal{F}}_t$ gives the usual time-indexed form. The denominator is the likelihood weight after reducing the full information to $\mathcal{G}$, not the original pointwise weight $Z$.