Let $P_1,P_2$ be probability measures on the measurable space $(\Omega,\mathfrak{G})$ on $\mathbb{R}^d$ and let $Y$ be data generated from either $P_i;i=1,2$.
Then the likelihoods satisfy
\begin{equation}
L(P_1|Y)>L(P_2|Y),
\label{eq_1}
\end{equation}
if and only if $Y$ was generated from $P_1$ and not from $P_2$ right?
My question is then, if $\mathfrak{F}$ is a sub-$\sigma$-algebra of $\mathfrak{G}$, then does $$ L(P_1(-|\mathfrak{F})|Y)>L(P_2(-|\mathfrak{F})|Y), $$ hold if and only if the first equation holds? Here $P_i(-|\mathfrak{F})$ is the conditional measure $P_i$ given the sub-$\sigma$-algebra $\mathfrak{F}$ of $\mathfrak{G}$.