Let ( , F , F , P ) ( , F , F , P ) be a filtered probability space and

Let $(\Omega ,F,F,P)$ be a filtered probability space and denote by $(L_t,t\ge 0)$ the Radon-Nikodým density of $Q$ with respect to $P$. Then, if $\tilde{F}$ is a subfiltration of $F$, prove that ${Q|}_{{\tilde{F}}_t}={{\tilde{L}}_{t}P|}_{{\tilde{F}}_t}$, where ${\tilde{L}}_t=E_P(L_t\mid {\tilde{F}}_t).$