= Bayes formula for conditional expectation
Suppose $\widetilde{\mathbb P}$ is a <probability measure> with $d\widetilde{\mathbb P}=Z\,d\mathbb P$, so $Z\geq0$ and $\mathbb EZ=1$. For a <sub-sigma-algebra> $\mathcal G$, write $Z_{\mathcal G}=\mathbb E[Z\mid\mathcal G]$. If $X$ is <integrable function>[integrable] under $\widetilde{\mathbb P}$, then on $\{Z_{\mathcal G}>0\}$,
$$
\widetilde{\mathbb E}[X\mid\mathcal G]=\frac{\mathbb E[ZX\mid\mathcal G]}{Z_{\mathcal G}}.
$$
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 $\widetilde{\mathbb P}$-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 $\int_A Y\,d\widetilde{\mathbb P}=\mathbb E[\mathbf1_AZY]=\mathbb E[\mathbf1_AZ_{\mathcal G}Y]=\mathbb E[\mathbf1_AZX]=\int_A X\,d\widetilde{\mathbb P}$. These are exactly the defining <conditional expectation>[conditional-expectation] identities. For a <filtration (probability theory)>[filtration], choosing $\mathcal G=\mathcal F_t$ gives the usual time-indexed form. The denominator is the <likelihood function>[likelihood] weight after reducing the full information to $\mathcal G$, not the original pointwise weight $Z$.
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