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

• Null empty set: The empty set has zero inner measure (see also: measure zero); that is, [math]\displaystyle{ \varphi(\varnothing) = 0 }[/math]
• Superadditive: For any disjoint sets [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B, }[/math] [math]\displaystyle{ \varphi(A \cup B) \geq \varphi(A) + \varphi(B). }[/math]
• Limits of decreasing towers: For any sequence [math]\displaystyle{ A_1, A_2, \ldots }[/math] of sets such that [math]\displaystyle{ A_j \supseteq A_{j+1} }[/math] for each [math]\displaystyle{ j }[/math] and [math]\displaystyle{ \varphi(A_1) \lt \infty }[/math] [math]\displaystyle{ \varphi \left(\bigcap_{j=1}^\infty A_j\right) = \lim_{j \to \infty} \varphi(A_j) }[/math]
• Infinity must be approached: If [math]\displaystyle{ \varphi(A) = \infty }[/math] for a set [math]\displaystyle{ A }[/math] then for every positive real number [math]\displaystyle{ r, }[/math] there exists some [math]\displaystyle{ B \subseteq A }[/math] such that [math]\displaystyle{ r \leq \varphi(B) \lt \infty. }[/math]

The inner measure induced by a measure

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

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

1. [math]\displaystyle{ \mu_*(\varnothing) = 0, }[/math]
2. [math]\displaystyle{ \mu_* }[/math] is non-negative,
3. If [math]\displaystyle{ E \subseteq F }[/math] then [math]\displaystyle{ \mu_*(E) \leq \mu_*(F). }[/math]

Measure completion

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

References

• 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)