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Inner measure

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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

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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

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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:

  1. is non-negative,
  2. If then

Measure completion

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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

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  • Lebesgue measurable set – Concept of area in any dimension

References

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  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)