I have taken an introductory course on measure theory where I learned about the Borel-Cantelli theorem but I wonder whether there is a lebesgue integrable version. Given an uncountable collection of independent events $E_{t \in \mathbb{R}_+}$,

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

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