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
[edit]An inner measure is a set function defined on all subsets of a set that satisfies the following conditions:
- Null empty set: The empty set has zero inner measure (see also: measure zero); that is,
- Superadditive: For any disjoint sets and
- Limits of decreasing towers: For any sequence of sets such that for each and
- If the measure is not finite, that is, if there exist sets with , then this infinity must be approached. More precisely, if for a set then for every positive real number there exists some such that
The inner measure induced by a measure
[edit]Let be a σ-algebra over a set and be a measure on Then the inner measure induced by is defined by
Essentially gives a lower bound of the size of any set by ensuring it is at least as big as the -measure of any of its -measurable subsets. Even though the set function is usually not a measure, shares the following properties with measures:
- is non-negative,
- If then
Measure completion
[edit]Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If is a finite measure defined on a σ-algebra over and and are corresponding induced outer and inner measures, then the sets such that form a σ-algebra with .[1] The set function defined by for all is a measure on known as the completion of
See also
[edit]- Lebesgue measurable set – Concept of area in any dimension
References
[edit]- ^ 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)