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 , we are interested in the following propositions:
Both statements are generally false. I’ll start with and then proceed with :
1. It can be shown that for any continuous which is positive for some index , there is an open interval such that:
From this it follows that there exists an increasing sequence such that:
Now, there are uncountably many non-negative continuous functions bounded by that are positive somewhere in whose integral converges. Ex:
Alternatively, note that if our collection of events is independent and has probability greater than some on a countable subset of the non-negative reals and has probability zero everywhere else then is integrable and its integral is zero over yet by the original Borel-Cantelli theorem
2. The second statement is also false without assuming independence of events. However, if we assume independence of events we have:
Using , we note that:
Assuming that is defined and Lebesgue-integrable we note that is closed and bounded so , we can define:
As a result we have:
By the Borel-Cantelli theorem, combined with 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 .