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 $E_{t \in \mathbb{R}_+}$,

$$ \int_0^{\infty} P(E_t) dt <\infty \implies P( E_t\quad i.o. )=0\tag{1}$$

$$ \int_0^{\infty} P(E_t) dt =\infty \implies P( E_t\quad i.o. )=1\tag{2}$$

Note: i.o. here means infinitely often. I'd like to add that this question is motivated by a problem I encountered in statistical physics.