User talk:Zaglabarg

Let the discussion commence!

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

Proof
Let (Ak) be a sequence of events in some probability space and suppose that the sum of the probabilities of the Ak 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."