Outer Measure of Limit of Increasing Sequence of Sets

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\mu^*$ be an outer measure on a set $X$.

Let $\left\langle{S_n}\right\rangle$ be an increasing sequence of $\mu^*$-measurable sets, and let $S_n \uparrow S$ (as $n \to \infty$).


Then for any subset $A \subseteq X$:

$\displaystyle \mu^* \left({A \cap S}\right) = \lim_{n \to \infty} \mu^* \left({A \cap S_n}\right)$


Proof

By the monotonicity of $\mu^*$, it suffices to prove that:

$\displaystyle \mu^* \left({A \cap S}\right) \le \lim_{n \mathop \to \infty} \mu^* \left({A \cap S_n}\right)$

Assume that $\mu^* \left({A \cap S_n}\right)$ is finite for all $n \in \N$, otherwise the statement is trivial by the monotonicity of $\mu^*$.


Let $S_0 = \varnothing$.

Then $x \in S$ if and only if there exists an integer $n \ge 0$ such that $x \in S_{n + 1}$.

Taking the least possible $n$, it follows that $x \notin S_n$, and so:

$x \in S_{n + 1} \setminus S_n$

Therefore:

$\displaystyle S = \bigcup_{n \mathop = 0}^\infty \left({S_{n + 1} \setminus S_n}\right)$

From Intersection Distributes over Union:

$\displaystyle A \cap S = A \cap \bigcup_{n \mathop = 0}^\infty \left({S_{n + 1} \setminus S_n}\right) = \bigcup_{n \mathop = 0}^\infty \left({A \cap \left({S_{n + 1} \setminus S_n}\right)}\right)$

Therefore:

\(\displaystyle \mu^* \left({A \cap S}\right)\) \(\le\) \(\displaystyle \sum_{n \mathop = 0}^\infty \mu^* \left({A \cap \left({S_{n + 1} \setminus S_n}\right)}\right)\) Definition of Countably Subadditive Function
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \left({\mu^* \left({A \cap S_{n + 1} }\right) - \mu^* \left({A \cap S_{n + 1} \cap S_n}\right)}\right)\) Definition of Measurability of $S_n$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \left({\mu^* \left({A \cap S_{n + 1} }\right) - \mu^* \left({A \cap S_n}\right)}\right)\) Intersection with Subset is Subset
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \mu^* \left({A \cap S_n}\right) - \mu^* \left({A \cap \varnothing}\right)\) Telescoping Series
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \mu^* \left({A \cap S_n}\right)\) Intersection with Empty Set and Definition of Outer Measure

$\blacksquare$