Why Must the Set of Points Belonging Infinitely to A_k Be a Null Set?

[complexnumber]

Homework Statement

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.

Homework Equations

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.

Is this correct?

 

[mighty2000]

Yes, your proof is correct. You have correctly used the fact that if the measure of a set is positive, then the measure of any subset of that set must also be positive. This leads to a contradiction with the given assumption that the sum of the measures of the sets A_k is finite. Therefore, the set S must have measure 0.