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.