I'm stuck on the exercise 1.17 of ch. VIII in Revuz and Yor. We have a filtration $(\mathcal{F}_t)$ and two probability measures $P,Q$ on $\mathcal{F}_\infty$.
Suppose that $Q = D_t P$ on $\mathcal{F}_t$ for a positive continuous martingale $D$. Prove that if $P,Q$ are mutually singular on $\mathcal{F}_\infty$, then $Y = \min_t D_t$ is uniformly distributed on $[0,1]$ under $Q$.
My idea was to introduction the stopping time $\tau_u = \inf \{t \geqslant 0 : D_t = u \}$, so that $Q(Y \leqslant u) = Q(\tau_u < \infty) = \mathbb{E}|\mathbf{1}_{\tau_u < \infty} D_{\tau_u}] = u P(\tau_u < \infty)$ (using proposition 1.3), but then ??
I can't see how to use the mutually singular assumption.