Inner measure

In mathematics, let S be a σ-algebra and μ be a measure on S. Following the notation in the text by Halmos, referenced below, let H(S) be the σ-algebra of all sets X such that there exists a set E $in$ S with X $subseteq$ E; i.e. H(S) is the set of all subsets of members of S. The inner measure μ_* on H(S) induced by μ is defined by

:μ_*(X) = sup{μ(F): F $in$ S and F $subseteq$ X }

The set function μ_* is usually not a measure on H(S). μ_* shares the following properties with measures::# μ_*($emptyset$)=0,:# μ_* is non-negative,:# If E$subseteq$ F then μ_*(E)$leq$μ_*(F).

Inner measures are often used in combination with outer measures to extend a measure μ defined on a σ-algebra S to a larger σ-algebra. If μ* is the outer measure induced by μ on H(S) then the sets X $in$ H(S) such that μ_*(X) = μ*(X) form a σ-algebra Ŝ with S $subseteq$ Ŝ $subseteq$ H(S). The set function μ^ defined by

:μ^(X) = μ*(X) = μ_*(X) for X $in$ Ŝ

is a measure on Ŝ known as the completion of μ.

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)