# Integrable variant of Borel-Cantelli theorem

In my measure theory course I was taught the Borel-Cantelli theorem but I wasn’t taught a Lebesgue-integrable variant. My previous blog post hints that this might be helpful for dealing with an uncountable collection of events in statistical physics. That’s the motivation and now I may proceed with my demonstration.

Given an uncountable collection of events $E_{t \in \mathbb{R}_+}$, we are interested in the following propositions:

$\int_0^{\infty} P(E_t) dt <\infty \implies P( E_t\quad i.o. )=0\quad (1)$
$\int_0^{\infty} P(E_t) dt =\infty \implies P( E_t\quad i.o. )=1\quad (2)$

Both statements are generally false. I’ll start with $(1)$ and then proceed with $(2)$:

1. It can be shown that for any continuous $P:t \rightarrow P(E_t)$ which is positive for some index $x \in \mathbb{R}_+$, there is an open interval $(a,b)$ such that:

$\exists N \in \mathbb{N}\forall t \in \mathbb{Q} \cap (a,b),P(E_t)>\frac{1}{N} \quad (3)$

From this it follows that there exists an increasing sequence ${\{q_n}\}_{n=0}^{\infty} \subset \mathbb{Q} \cap (a,b)$ such that:

$\sum_{n=0}^{\infty} P(E_{q_n}) = \infty \quad(4)$

Now, there are uncountably many non-negative continuous functions bounded by $1$ that are positive somewhere in $\mathbb{R}_+$ whose integral converges. Ex: $f(x) = e^{-x}$

Alternatively, note that if our collection of events is independent and has probability greater than some $\epsilon > 0$ on a countable subset of the non-negative reals and has probability zero everywhere else then $P(E_t)$ is integrable and its integral is zero over $\mathbb{R}_+$ yet by the original Borel-Cantelli theorem $P(E_t \quad i.o.) =1$

2. The second statement is also false without assuming independence of events. However, if we assume independence of events we have:

$\{{q_n}\}_{n=0}^{\infty} \subset \mathbb{R}_+ \implies \{E_{q_n}\}_{n=0}^{\infty} \subset {E_{\mu \in \mathbb{R}_+}} \quad (5)$

Using $(5)$, we note that:

$P(E_{q_n} \quad i.o.)=1 \implies P( E_\mu \quad i.o. )=1\quad (6)$

Assuming that $P(E_\mu)$ is defined $\forall \mu \in \mathbb{R}_+$ and Lebesgue-integrable we note that $\{P(E_\mu):\mu\in[n,n+1-\frac{1}{2^n}]\subset \mathbb{N}\}$ is closed and bounded so $\forall n \in \mathbb{N}$, we can define:

$P(E_{q_n}) =\sup\limits_{\mu\in [n,n+1 -\frac{1}{2^n}]}{P(E_\mu)} \quad (7)$

As a result we have:

$\int_0^{\infty} P(E_\mu) d\mu =\infty \implies \sum_{n=0}^{\infty} P(E_{q_n})=\infty \quad (8)$

By the Borel-Cantelli theorem, $(8)$ combined with $(6)$ gives us the desired result.

Note: It may be possible to argue that events that are spaced by sufficiently large time intervals are so weakly-correlated that they are practically independent. In that case it might be possible to use the argument I used for $(2)$.