Measure is Countably Subadditive
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Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Then $\mu$ is a countably subadditive function.
Proof
Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence of sets in $\Sigma$.
It is required to show that:
- $\ds \map \mu {\bigcup_{n \mathop \in \N} E_n} \le \sum_{n \mathop \in \N} \map \mu {E_n}$
Now define the sequence $\sequence {F_n}_{n \mathop \in \N}$ in $\Sigma$ by:
- $F_n := \ds \bigcup_{k \mathop = 0}^n E_n$
By Subset of Union, it follows that, for all $n \in \N$, $F_n \subseteq F_{n + 1}$.
Hence, $\sequence {F_n}_{n \mathop \in \N}$ is increasing.
It is immediate that $F_n \uparrow \ds \bigcup_{n \mathop \in \N} E_n$, where $\uparrow$ signifies the limit of an increasing sequence of sets.
Now reason as follows:
\(\ds \map \mu {\bigcup_{n \mathop \in \N} E_n}\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \map \mu {F_n}\) | Measure of Limit of Increasing Sequence of Measurable Sets | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \map \mu {E_0 \cup \cdots \cup E_n}\) | Definition of $F_n$ | |||||||||||
\(\ds \) | \(\le\) | \(\ds \lim_{n \mathop \to \infty} \sum_{k \mathop = 0}^n \map \mu {E_k}\) | Measure is Subadditive: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \in \N} \map \mu {E_k}\) |
Hence the result.
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.6$