Exercise 1.21 (Counterexample to the relaxed Borel-Cantelli lemma)

Question

Let $p_1,p_2,\dots \in [0,1]$ 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.

Elu Thingol · Accepted Answer

Consider $([0,1],\mathcal{B},m)$ and the sequence $([0,p_n])_{n\in\mathbb{N}}$. Since we have infinitely many $p_n > 0$ we must as well have infinitely many non-null sets $[0,p_n]$.\par

Also see part 2 of \href{../an_introduction_to_measure_theory/exercise_1-4-44}{this exercise}.

Exercise 1.21 (Counterexample to the relaxed Borel-Cantelli lemma)

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Exercise 1.21 (Counterexample to the relaxed Borel-Cantelli lemma)

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