Absolute continuity and singularity of two probability measures on a filtered space

Let $\mu$ and $\nu$ be fixed probability measures on a filtered space $(\Omega, {\cal F}, ({\cal F}_t)_{t\in {\bf R}^{+}})$. Denote by $\mu_T$ and $\nu_T$ (respectively, $\mu_{T-}$ and $\nu_{T-}$) the restrictions of the measures $\mu$ and $\nu$ on ${\cal F}_T$ (respectively, on ${\cal F}_{T-}$) for a stopping time $T$. We find the Hahn decomposition of $\mu_T$ and $\nu_T$ using the Hahn decomposition of the measures $\mu$, $\nu$, and the Hellinger process $h_t$ in the strict sense of order 1/2. The norm of the absolutely continuous component of $\mu_{T-}$ with respect to $\nu_{T-}$ is computed in terms of density processes and Hellinger integrals.