Outer measure

Definition. Let $X$ be a nonempty set. An outer measure on the set $X$ is a function $\mu ^{*}:2^{X}\to [0,\infty ]$ such that

• $\mu ^{*}(\emptyset )=0$,
• $\mu ^{*}(A)\leq \mu ^{*}(B)$ if $A\subseteq B$,
• $\mu ^{*}\left(\bigcup \limits _{j=1}^{\infty }A_{j}\right)\leq \sum _{j=1}^{\infty }\mu ^{*}(A_{j}).$

The second and third conditions in the definition of an outer measure are equivalent to the condition that $A\subseteq \bigcup \limits _{i=1}^{\infty }B_{i}$ implies $\mu ^{*}(A)\leq \sum _{i=1}^{\infty }\mu ^{*}(B_{i})$.

Definition. A set $A\subset X$ is called Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu^* } -measurable if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu^*(E) = \mu^*(E \cap A) + \mu^* (E \cap A^c)} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E \subset X } .

Examples of Outer Measures

The standard example of an outer measure is the Lebesgue outer measure, defined on subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu^*(A) = \inf \left\{ \sum_{i=1}^\infty |b_i - a_i| : A \subseteq \bigcup_{i=1}^\infty (a_i, b_i) \right\}. }

A near-generalization of the Lebesgue outer measure is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu^*_F(A) = \inf \left\{ \sum_{i=1}^\infty |F(b_i) - F(a_i)| : A \subseteq \bigcup_{i=1}^\infty (a_i, b_i] \right\}, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is any right-continuous function .

Given a measure space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X, \mathcal{M}, \mu)} , one can always define an outer measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu^*} by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu^*(A) = \inf \left\{ \mu(B) : A \subseteq B, B \in \mathcal{M} \right\}. }