I have taken an introductory course on measure theory where I learned about the Borel-Cantelli theorem but I wonder whether there is a continuous version. Given $E_t$ with $t>0$, 

$$ \int_0^{\infty} P(E_t) dt =\infty \implies P\{ E_t \quad i.o. \}=1\tag{*}$$ 

The question occurred to me when I considered a problem from statistical physics: http://math.stackexchange.com/questions/2077097/microcanonical-distribution