Measure is Subadditive/Corollary
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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.
$\blacksquare$