Radon-Nikodym density martingale ($Z_t={\mathbb{E}}_{\mathbb{P}}[d\mathbb{Q}/d\mathbb{P}\mid {\mathcal{F}}_t]$)

For an absolutely continuous change of probability measure, restricting its Radon-Nikodym derivative to the current sigma-algebra gives this nonnegative uniformly integrable martingale. If $Z_T>0$ almost surely, the measures are equivalent on ${\mathcal{F}}_T$. This finite-horizon equivalence need not imply equivalence on a larger sigma-algebra.