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) .


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