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.

Definition

An inner measure is a function

${\displaystyle \varphi :2^{X}\rightarrow [0,\infty ],}$

defined on all subsets of a set X, that satisfies the following conditions:

• Null empty set: The empty set has zero inner measure (see also: measure zero).

${\displaystyle \varphi (\varnothing )=0}$

• Superadditive: For any disjoint sets A and B,

${\displaystyle \varphi (A\cup B)\geq \varphi (A)+\varphi (B).}$

• Limits of decreasing towers: For any sequence {A_j} of sets such that ${\displaystyle A_{j}\supseteq A_{j+1}}$ for each j and ${\displaystyle \varphi (A_{1})<\infty }$

${\displaystyle \varphi \left(\bigcap _{j=1}^{\infty }A_{j}\right)=\lim _{j\to \infty }\varphi (A_{j})}$

• Infinity must be approached: If ${\displaystyle \varphi (A)=\infty }$ for a set A then for every positive real number c, there exists B ⊆ A such that,

${\displaystyle c\leq \varphi (B)<\infty }$

The inner measure induced by a measure

Let Σ be a σ-algebra over a set X and μ be a measure on Σ. Then the inner measure μ_* induced by μ is defined by

${\displaystyle \mu _{*}(T)=\sup\{\mu (S):S\in \Sigma {\text{ and }}S\subseteq T\}.}$

Essentially μ_* gives a lower bound of the size of any set by ensuring it is at least as big as the μ-measure of any of its Σ-measurable subsets. Even though the set function μ_* is usually not a measure, μ_* shares the following properties with measures:

1. μ_*(∅) = 0,
2. μ_* is non-negative,
3. If E ⊆ F then μ_*(E) ≤ μ_*(F).

Measure completion

Main article: complete measure

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

${\displaystyle {\hat {\mu }}(T)=\mu ^{*}(T)=\mu _{*}(T)}$

for all ${\displaystyle T\in {\hat {\Sigma }}}$ is a measure on ${\displaystyle {\hat {\Sigma }}}$ known as the completion of μ.

References

1. Halmos 1950, § 14, Theorem F

• 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)