Conditional expectation under a martingale density (${\mathbb{E}}_Q[Y\mid {\mathcal{F}}_s]={\mathbb{E}}_P[Y{Z}_T\mid {\mathcal{F}}_s]/Z_s$)

Let $Z_s>0$ be a martingale with $Z_0=1$ and let $dQ=Z_{T}dP$ on a finite horizon. Its restriction to ${\mathcal{F}}_s$ has Radon-Nikodym derivative $Z_s$. If $Y$ is Q-integrable and $U={\mathbb{E}}_Q[Y\mid {\mathcal{F}}_s]$, then for every $A\in {\mathcal{F}}_s$,Also $Z_{s}U$ is P-integrable because ${\mathbb{E}}_P[Z_s|U|]={\mathbb{E}}_Q|U|<\infty$. The defining uniqueness of conditional expectation gives the displayed formula. In particular, if $N_s{Z}_s$ is a P-martingale and $N_s$ is Q-integrable, conditioning first at an intermediate time proves $N_s$ is a Q-martingale.