Given a sequence of events $A_n$ for $n\in \mathbb N$, the first Borel Cantelli lemma states that, if the sum over all probabilities $\sum_{i=1}^n P(A_n)$ is finite, then the probability of the limit supremum of the $A_n$ is zero. Or more intuitively, if the sum $\sum_{i=1}^n P(A_n)$ is finite, then the probability for $\{ A_n$ happens infinitely often } is zero.
Now my question is: is there a continuous version of this?
i.e. is there any statement like: given $A_t$ with $t>0$, if $ \int_0^{\infty} P(A_t) dt<\infty$ then $P\{ A_t$ happens infinitely often }$=0$?
Best Answer
Let $T$ be the collection of all the $t$ that $A_t$ happens. That is, for any $\omega$, an element in the probability space which your are working with, define $$ T(\omega) := \{t\geq 0: \mathbf 1_{A_t} (\omega)=1\}. $$ Denote $l(T)$ as the Lebesgue measure of this random set $T$. Then, by Fubini's theorem, $$ \int_0^\infty \mathbf P[A_t]dt = \mathbf E[l(T)]. $$ Therefore, if the left side is finite, we surely have that $$ l(T)<\infty $$ happens as surely.