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 defined on all subsets that satisfies the following conditions:

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

• Superadditive: For any disjoint

• Limits of decreasing towers: For any sequence

• then for every positive real number

The inner measure induced by a measure

be a σ-algebra be a measure

shares the following properties with measures:

1. is non-negative,

Measure completion

Induced inner measures are often used in combination with outer measures is a finite measure defined on a σ-algebra.

See also

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, (Chapter 7)