Let $(\Omega,\mathscr{F}, P)$ be a probability space and let $Q$ be another probability measure on $\mathscr{F}$, and let $\mathscr{F}_n=\sigma(Y_1,\ldots,Y_n)$ be a non-decreasing sequence of $\sigma$-fields in $\mathscr{F}$, for random variables $Y_1,Y_2,\ldots$ on $(\Omega,\mathscr{F})$.
Suppose that the distribution of the random vector $(Y_1,\ldots,Y_n)$ has densities $p_n(y_1,\ldots,y_n)$ and $q_n(y_1,\ldots,y_n)$, with respect to $n$-dimensional Lebesgue measure, under the measures $P$ and $Q$ respectively.
Now, suppose the $Y_n$ are independent and identically distributed under both $P$ and $Q$.
(So, for example, $p_n(y_1,\ldots,y_n)=p(y_1)\cdots p(y_n)$ for density $p$ on the line).
If $P[Y_n \in H]\neq Q[Y_n\in H]$ for some $H \in \mathscr{R}^1$ and $Z_n=I_{[Y_n\in H]}$, then what happens to $n^{-1}\sum_{k=1}^n Z_k$ by the strong law of large numbers?
I mean, does it converge to $P[Y_1 \in H]$ or to $Q[Y_1 \in H]$?
Additionally, if $P$ dominates $Q$ when both are restricted to $\mathscr{F}_n$, can $P$ and $Q$ be mutually singular on $\mathscr{F}$ and how?
Thanks and Regards!