probability of a sequence of event and their intersection with a an event whose liminf has probabilty 1

Question

Let $P_n=\times_{i=1}^n P$ denote a sequence of product probability measures and $E_n$ a sequence of measurable events such that $P_\infty(\liminf_{n\to \infty}E_n)=1$. Can we conclude that for any other sequence or measurable events $S_n$ we have $$ \lim_{n \to \infty} \frac{P_n(E_n \cap S_n)}{P_n(S_n)}=1 $$ or not?

Answer 1

Would this be a counter-example? Apologies if I misunderstood the question.

• $\forall n: E_n \cap S_n = \emptyset$ so the numerator is always $0$.

• $\forall n: P_n(S_n) > 0$ so the denominator is always positive.

• $\lim_{n \to \infty} P_n(S_n) = 0$

I am not exactly sure how to interpret $P_\infty (\lim \inf_{n \to \infty} E_n) = 1$ but the last bullet above should hopefully make it possible...? Anyway, if all $3$ bullets can be satisfied then the ratio is always $0$ and its limit is also $0$.