proof of Borel-Cantelli 1

Let Bk be the event i=kAi for k=1,2,,. If x is in the event Ai’s i.o., then xBk for all k. So xk=1Bk.

Conversely, if xBk for all k, then we can show that x is in Ai’s i.o. Indeed, xB1=i=1Ai means that xAj(1) for some j(1). However xBj(1)+1 implies that xAj(2) for some j(2) that is strictly larger than j(1). Thus we can produce an infiniteMathworldPlanetmath sequence of integer j(1)<j(2)<j(3)< such that xAj(i) for all i.

Let E be the event {x:xAi i.o.}. We have


From EBk for all k, it follows that P(E)P(Bk) for all k. By union bound, we know that P(Bk)i=kP(Ai). So P(Bk)0, by the hypothesisMathworldPlanetmathPlanetmath that i=1P(Ai) is finite. Therefore, P(E)=0.

Title proof of Borel-Cantelli 1
Canonical name ProofOfBorelCantelli1
Date of creation 2013-03-22 14:28:16
Last modified on 2013-03-22 14:28:16
Owner kshum (5987)
Last modified by kshum (5987)
Numerical id 6
Author kshum (5987)
Entry type Proof
Classification msc 60A99