Skip to content

Outer Measures#

Definition: Outer Measure

Let \(X\) be a set.

An outer measure on \(X\) is a function \(\mu^{\ast}: \mathcal{P}(X) \to \mathbb{R}_{\ge 0} \cup \{\infty\}\) from the powerset of \(X\) to the non-negative extended real numbers with the following properties:

  • \(\mu^{\ast}(\varnothing) = 0\)
  • For all subsets \(A\) and \(B\) of \(X\), we have
\[ A \subseteq B \implies \mu^{\ast}(A) \le \mu^{\ast}(B) \]
\[ \mu^{\ast} \left(\bigcup_{i=1}^{\infty} S_i \right) \le \sum_{i = 1}^{\infty} \mu^{\ast}(S_i) \]

Notation

We usually write \(\mathbb{R}_{\ge 0} \cup \{\infty\}\) as \([0, \infty]\).

Warning

An outer measure is not necessarily a measure on \((X, \mathcal{P}(X))\).

Theorem: Alternative Definition

Let \(X\) be a set.

A function \(\mu^{\ast}: \mathcal{P}(X) \to \mathbb{R}_{\ge 0} \cup \{\infty\}\) from the powerset of \(X\) to the non-negative extended real numbers is an outer measure on \(X\) if and only if \(\mu^{\ast}\) has the following properties:

  • \(\mu^{\ast}(\varnothing) = 0\)
  • For all subsets \(A \subseteq X\) and all countable subcollections \(\{S_i\}_{i = 1}^{\infty}\) of \(\mathcal{P}(X)\) we have
\[ A \subseteq \bigcup_{i = 1}^{\infty} S_i \implies \mu^{\ast}(A) \le \sum_{i = 1}^{\infty} \mu^{\ast}(S_i) \]
Proof

Proof of (1):

TODO

Proof of (2):

TODO

Definition: Carathéodory Measurability

Let \(X\) be a set with an outer measure \(\mu^{\ast}\).

A subset \(E \subseteq X\) is Carathéodory-measurable relative to \(\mu^{\ast}\) or \(\mu^{\ast}\)-measurable if

\[ \mu^{\ast}(S) = \mu^{\ast}(S \cap E) + \mu^{\ast} (S \setminus E) \]

for every subset \(S \subseteq X\).

Carathéodory's Extension Theorem

If \(X\) is a set with an outer measure \(\mu^{\ast}\), then:

Proof

TODO