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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.

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  • $\begingroup$ As written, this must be false: for example, let all $E_t$ be the same event, with probability $1/2$. Then $P(E_t \, i.o.)=P(E_1)=1/2$. You would need to add some condition of independence, or decay of correlations, or the like... $\endgroup$
    – James Martin
    Commented Dec 30, 2016 at 12:02
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    $\begingroup$ "Clarified" as in "changed" :) Now it is implied by the normal Borel-Cantelli lemma, since you can find a countable sequence $t_k$ such that $\sum P(E_{t_k})=\infty$. $\endgroup$
    – James Martin
    Commented Dec 30, 2016 at 15:54
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    $\begingroup$ Both of these now look wrong. For the second, aren't we back where we started? Without some condition of independence, or correlation decay, this one must be doomed. For the first, let $E_t$ be independent with $P(E_t)=1$ if $t$ is an integer, and $P(E_t)=0$ otherwise. Then the integral is zero, but from the standard B-C lemma, w.p.1 there will be infinitely many integers $t$ such that $E_t$ occurs. Or, alternatively, let $E_t$ be independent with $P(E_t)=e^{-t}$. Then the integral is finite, but for any $0<a<b<1$ there are uncountably many events $E_t$ with probability in $(a,b)$. $\endgroup$
    – James Martin
    Commented Dec 31, 2016 at 18:22
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    $\begingroup$ Which actually raises another issue with this question: given an uncountable family of events $E_t$, the set $\{E_t \text{ i.o.}\}$ need not be measurable, so without more assumptions it does not even make sense to talk about its probability. $\endgroup$
    – Nate Eldredge
    Commented Apr 23, 2019 at 15:33
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    $\begingroup$ I don't know a book that studies this issue explicitly and in depth, but you see examples of the phenomenon in many places. For instance, I like the construction of Brownian motion in Durrett's Probability: Theory and Examples, which really only constructs it at a countable dense set of times, and shows that it is uniformly continuous on this set. Other books try to really construct an uncountable family of random variables and end up with awkward steps involving "modifications" of the process. $\endgroup$
    – Nate Eldredge
    Commented Apr 23, 2019 at 20:15

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The second Borel-Cantelli lemma has the additional condition that the events are mutually independent. This requirement becomes problematic for an uncountable index set. For example, suppose that $\{E_t\}_{0\le t\le 1}$ is a collection of mutally independent events such that $(\omega,t)\mapsto 1_{E_t}(\omega)$ is measurable. (This is needed for the integrations you envision.) Then $$ \int_a^b 1_{E_t}\,dt =\int_a^bp(t)\,dt,\qquad a.s., $$ where $p(t):=P(E_t)$. (To see this compute the variance of $\int_a^b 1_{E_t}\,dt$.) From this and the Lebesgue density and differentiation theorems it follows that $P(E_t)$ is $0$ or $1$ for a.e. $t$. Thus mutual independence and measurability hold only in trivial situations.

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