Inner measure

In mathematics, in particular in measure theory, an inner measure is a function on the power set of a given set, with values in the extended real numbers, satisfying some technical conditions. Intuitively, the inner measure of a set is a lower bound of the size of that set.

An inner measure is a set function $${\displaystyle \varphi :2^{X}\to [0,\infty ],}$${\displaystyle \varphi :2^{X}\to [0,\infty ],} defined on all subsets of a set ${\displaystyle X,}${\displaystyle X,} that satisfies the following conditions:

• Null empty set: The empty set has zero inner measure (see also: measure zero ); that is, $${\displaystyle \varphi (\varnothing )=0}$${\displaystyle \varphi (\varnothing )=0}
• Superadditive: For any disjoint sets ${\displaystyle A}${\displaystyle A} and ${\displaystyle B,}${\displaystyle B,} $${\displaystyle \varphi (A\cup B)\geq \varphi (A)+\varphi (B).}$${\displaystyle \varphi (A\cup B)\geq \varphi (A)+\varphi (B).}
• Limits of decreasing towers: For any sequence ${\displaystyle A_{1},A_{2},\ldots }${\displaystyle A_{1},A_{2},\ldots } of sets such that ${\displaystyle A_{j}\supseteq A_{j+1}}${\displaystyle A_{j}\supseteq A_{j+1}} for each ${\displaystyle j}${\displaystyle j} and ${\displaystyle \varphi (A_{1})<\infty }${\displaystyle \varphi (A_{1})<\infty } $${\displaystyle \varphi \left(\bigcap _{j=1}^{\infty }A_{j}\right)=\lim _{j\to \infty }\varphi (A_{j})}$${\displaystyle \varphi \left(\bigcap _{j=1}^{\infty }A_{j}\right)=\lim _{j\to \infty }\varphi (A_{j})}
• If the measure is not finite, that is, if there exist sets ${\displaystyle A}${\displaystyle A} with ${\displaystyle \varphi (A)=\infty }${\displaystyle \varphi (A)=\infty }, then this infinity must be approached. More precisely, if ${\displaystyle \varphi (A)=\infty }${\displaystyle \varphi (A)=\infty } for a set ${\displaystyle A}${\displaystyle A} then for every positive real number ${\displaystyle r,}${\displaystyle r,} there exists some ${\displaystyle B\subseteq A}${\displaystyle B\subseteq A} such that $${\displaystyle r\leq \varphi (B)<\infty .}$${\displaystyle r\leq \varphi (B)<\infty .}

Let ${\displaystyle \Sigma }${\displaystyle \Sigma } be a σ-algebra over a set ${\displaystyle X}${\displaystyle X} and ${\displaystyle \mu }${\displaystyle \mu } be a measure on ${\displaystyle \Sigma .}${\displaystyle \Sigma .} Then the inner measure ${\displaystyle \mu _{*}}${\displaystyle \mu _{*}} induced by ${\displaystyle \mu }${\displaystyle \mu } is defined by $${\displaystyle \mu _{*}(T)=\sup\{\mu (S):S\in \Sigma {\text{ and }}S\subseteq T\}.}$${\displaystyle \mu _{*}(T)=\sup\{\mu (S):S\in \Sigma {\text{ and }}S\subseteq T\}.}

Essentially ${\displaystyle \mu _{*}}${\displaystyle \mu _{*}} gives a lower bound of the size of any set by ensuring it is at least as big as the ${\displaystyle \mu }${\displaystyle \mu }-measure of any of its ${\displaystyle \Sigma }${\displaystyle \Sigma }-measurable subsets. Even though the set function ${\displaystyle \mu _{*}}${\displaystyle \mu _{*}} is usually not a measure, ${\displaystyle \mu _{*}}${\displaystyle \mu _{*}} shares the following properties with measures:

1. ${\displaystyle \mu _{*}(\varnothing )=0,}${\displaystyle \mu _{*}(\varnothing )=0,}
2. ${\displaystyle \mu _{*}}${\displaystyle \mu _{*}} is non-negative,
3. If ${\displaystyle E\subseteq F}${\displaystyle E\subseteq F} then ${\displaystyle \mu _{*}(E)\leq \mu _{*}(F).}${\displaystyle \mu _{*}(E)\leq \mu _{*}(F).}

Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If ${\displaystyle \mu }${\displaystyle \mu } is a finite measure defined on a σ-algebra ${\displaystyle \Sigma }${\displaystyle \Sigma } over ${\displaystyle X}${\displaystyle X} and ${\displaystyle \mu ^{*}}${\displaystyle \mu ^{*}} and ${\displaystyle \mu _{*}}${\displaystyle \mu _{*}} are corresponding induced outer and inner measures, then the sets ${\displaystyle T\in 2^{X}}${\displaystyle T\in 2^{X}} such that ${\displaystyle \mu _{*}(T)=\mu ^{*}(T)}${\displaystyle \mu _{*}(T)=\mu ^{*}(T)} form a σ-algebra ${\displaystyle {\hat {\Sigma }}}${\displaystyle {\hat {\Sigma }}} with ${\displaystyle \Sigma \subseteq {\hat {\Sigma }}}${\displaystyle \Sigma \subseteq {\hat {\Sigma }}}.^[1] The set function ${\displaystyle {\hat {\mu }}}${\displaystyle {\hat {\mu }}} defined by $${\displaystyle {\hat {\mu }}(T)=\mu ^{*}(T)=\mu _{*}(T)}$${\displaystyle {\hat {\mu }}(T)=\mu ^{*}(T)=\mu _{*}(T)} for all ${\displaystyle T\in {\hat {\Sigma }}}${\displaystyle T\in {\hat {\Sigma }}} is a measure on ${\displaystyle {\hat {\Sigma }}}${\displaystyle {\hat {\Sigma }}} known as the completion of ${\displaystyle \mu .}${\displaystyle \mu .}

• Lebesgue measurable set – Concept of area in any dimensionPages displaying short descriptions of redirect targets

• Halmos, Paul R., Measure Theory, D. Van Nostrand Company, Inc., 1950, pp. 58.
• A. N. Kolmogorov & S. V. Fomin, translated by Richard A. Silverman, Introductory Real Analysis, Dover Publications, New York, 1970, ISBN 0-486-61226-0 (Chapter 7)

• Measures (measure theory)

• Pages displaying short descriptions of redirect targets via Module:Annotated link