Zaglabarg

Let the discussion commence!

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http://en.wikipedia.org/wiki/File:Girsanov.png

Proof

Let (A_k) be a sequence of events in some probability space and suppose that the sum of the probabilities of the A_k is finite. That is suppose:

$\sum _{k=1}^{\infty }P(A_{k})<\infty .$

Note that the convergence of this sum implies:

$\inf _{m\geq 1}\sum _{k=m}^{\infty }P(A_{k})=0.\,$

Therefore it follows that:

$P\left(\limsup _{n\to \infty }A_{k}\right)=P(A_{k}{\text{ i.o.}})=P\left(\bigcap _{m=1}^{\infty }\bigcup _{k=m}^{\infty }A_{k}\right)\leq \inf _{m\geq 1}P\left(\bigcup _{k=m}^{\infty }A_{k}\right)\leq \inf _{m\geq 1}\sum _{k=m}^{\infty }P(A_{k})=0$

where the abbreviation "i.o." denotes "infinitely often."