Measure space, null set

1. May 24, 2010

complexnumber

1. The problem statement, all variables and given/known data

Let $$(X,\mathcal{A},\mu)$$ be a fixed measure space.

Let $$A_k \in \mathcal{A}$$ such that $$\displaystyle \sum^\infty_{k=1} \mu(A_k) < \infty$$. Prove that
\begin{align*} \{ x \in X | x \in A_k \text{ for infinitely many k} \} \end{align*}
is a null set.

2. Relevant equations

3. The attempt at a solution

Let $$S = \{ x \in X | x \in A_k \text{ for infinitely many k} \}$$.
Suppose $$\mu (S) > 0$$. Then $$\displaystyle \mu(\bigcap A_k) > 0, A_k \ni x, x \in S$$. Then $$\mu (A_k) > 0, A_k \ni x, x \in S$$ and hence
$$\displaystyle \sum^\infty_{k=1} \mu(A_k) = \infty$$, which
contradicts to assumption.

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