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## Borel Cantelli Lemma

Suppose $\{E_k\}_{k=1}^\infty$ is a countable family of measurable sets of a measure space $(X,\mathcal{A},m)$ and $\displaystyle \sum_{k=1}^\infty m(E_k)< \infty$.
Then if we define $E=\displaystyle \bigcap_{n=1}^\infty \bigcup_{k\geq n}E_k= \limsup_{k \to \infty}E_k$ we have $m(E)=0$.

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1. April 24, 2010 at 8:14 pm