 that for $\mathbb P$-a.e. $\omega\in\Omega$ and for $ds\times d\mathbb P^{\tau,\omega}$-a.e. $(s,\widetilde\omega)\in[\tau(\omega),T]\times\Omega^{\tau(\omega)}$ $$\int_\tau^s\left(\nu_r^{\mathbb P,\tau,\omega}(\widetilde\omega,A)-\nu^{\tau(\omega),\mathbb P^{\tau,\omega}}_r(\widetilde\omega,A)\right)dr