Measure is Subadditive/Corollary

Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $E_1, \ldots, E_n \in \Sigma$.

Then:

$\ds \map \mu {\bigcup_{k \mathop = 1}^n E_k} \le \sum_{k \mathop = 1}^n \map \mu {E_k}$.

Proof

We have Measure is Subadditive.

The result follows by an application of Finite Union of Sets in Subadditive Function.

$\blacksquare$