Measure is Subadditive/Corollary

Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

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

Then:

$\displaystyle \mu \left({\bigcup_{k \mathop = 1}^n E_k}\right) \le \sum_{k \mathop = 1}^n \mu \left({E_k}\right)$.

Proof

We have Measure is Subadditive.

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

$\blacksquare$