Relaxed Borel-Cantelli Lemma

Question

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

Answer 1

Start with the following easy lemma:

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.