- Let \(\{ E _ { k } \}\) be a sequence of (Lebesgue) measurable sets in \(\mathbb { R } ^ { d }\) such that
\[ \sum _ { k = 1 } ^ { \infty } m ( E _ { k } ) < \infty . \]
Prove that almost every \(\boldsymbol { x } \in \mathbb { R } ^ { d }\) lies in at most finitely many sets \(E _ { k }\).
Solution
1. Express the points lying in infinitely many sets as a limsup.
Let \[ E:=\limsup_{k\to\infty}E_k =\bigcap_{N=1}^\infty\bigcup_{k\ge N}E_k. \] A point \(x\) belongs to \(E\) exactly when it belongs to infinitely many of the sets \(E_k\).
2. Estimate the measure of every tail union.
For every \(N\), \[ E\subseteq\bigcup_{k\ge N}E_k. \] By countable subadditivity, \[ m(E) \le m\left(\bigcup_{k\ge N}E_k\right) \le\sum_{k\ge N}m(E_k). \] The series \[ \sum_{k=1}^\infty m(E_k) \] converges by hypothesis, so its tails tend to zero. Hence \[ m(E)=0. \] Therefore almost every point belongs to only finitely many of the sets \(E_k\).