# Relaxed Borel-Cantelli Lemma

Let $${p_1,p_2,\ldots \in [0,1]}$$ be a sequence such that $${\sum_{n=1}^\infty p_n = +\infty}$$. Show that there exist a sequence of events $${E_1,E_2,\dots}$$ modeled by some probability space $${\Omega}$$, such that $${{\bf P}(E_n)=p_n}$$ for all $${n}$$, and such that almost surely infinitely many of the $${E_n}$$ occur. Thus we see that the hypothesis $${\sum_{n=1}^\infty {\bf P}(E_n) < \infty}$$ in the Borel-Cantelli lemma cannot be relaxed.

At first I thought of the Borel $$\sigma$$-algebra of the unit interval with the lebesgue measure, then define $$E_n := [0, p_n]$$ for each $$n$$. But this seems not to work, any hint will be appreciated

• The conclusion of this does not make any sense!! Anyways, just take $X_i \sim \mathcal U[0,1]$ i.i.d and conclude with the second bc lemma. Commented Aug 22, 2023 at 1:32
• Use intervals like $[0,p_1], [p_1,p_1+p_2],[p_1+p_2,p_1+p_2+p_3], \dots$ where you wrap around from $1$ to $0$. That is, in $[0,1]$ identify the endpoints to get a circle. Commented Aug 22, 2023 at 1:46
• @Andrew This is part of a series of notes on introductory probability given in Tao' blog, this is Exercise 21 in terrytao.wordpress.com/2015/10/03/…. In particular, the second BC has not been derived at this point. Commented Aug 22, 2023 at 1:49
• @GEdgar: This is what I had in mind as well. i.e. If the sum of the probability is infinite, one should be able to cover $[0, 1]$ with arbitrary thickness. Commented Aug 22, 2023 at 1:51
• @KKslider: Please edit your question to cite the source of the problem. Commented Aug 22, 2023 at 17:45

Suppose we have finitely many probabilities $$p_1, \dots, p_N$$ with $$p_1 + \dots + p_N \ge 1$$. Then on some reasonable probability space such as $$[0,1]$$ (i.e. one $$U(0,1)$$ random variable), there exist events $$E_1, \dots, E_N$$ such that $$P(E_n) = p_n$$ and $$P\left(\bigcup_{n=1}^N E_n\right)=1$$. That is, almost surely, at least one of the $$E_n$$ happens.
Now given an infinite sequence with $$\sum p_n = +\infty$$, we can partition it into infinitely many finite sets, each of which has a sum of at least 1. Apply the lemma to each set and conclude.