Inner measure

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In mathematics, in particular in measure theory, an inner measure is a function on the set of all subsets 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 function

\varphi: 2^X \rightarrow [0, \infty],

defined on all subsets of a set X, that satisfies the following conditions:

 \varphi(\varnothing) = 0
 \varphi( A \cup B) \geq \varphi(A) + \varphi( B ).
  • Limits of decreasing towers: For any sequence {Aj} of sets such that  A_j \supseteq A_{j+1} for each j and  \varphi(A_1) < \infty
 \varphi \left(\bigcap_{j=1}^\infty A_j\right) = \lim_{j \to \infty} \varphi(A_j)
  • Infinity must be approached: If \varphi(A) = \infty for a set A then for every positive number c, there exists a B which is a subset of A such that,
 c \leq \varphi( B) <\infty

The inner measure induced by a measure[edit]

Let Σ be a σ-algebra over a set X and μ be a measure on Σ. Then the inner measure μ* induced by μ is defined by

\mu_*(T)=\sup\{\mu(S):S\in\Sigma\text{ and }S\subseteq T\}.

Essentially μ* gives a lower bound of the size of any set by ensuring it is at least as big as the μ-measure of any of its Σ-measurable subsets. Even though the set function μ* is usually not a measure, μ* shares the following properties with measures:

  1. μ*(∅)=0,
  2. μ* is non-negative,
  3. If EF then μ*(E) ≤ μ*(F).

Measure completion[edit]

Main article: complete measure

Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If μ is a measure defined on σ-algebra Σ over X and μ* and μ* are corresponding induced outer and inner measures, then the sets T ∈ 2X such that μ*(T) = μ*(T) form a σ-algebra \scriptstyle\hat\Sigma with \scriptstyle\Sigma\subseteq\hat\Sigma. The set function μ̂ defined by

\hat\mu(T)=\mu^*(T)=\mu_*(T),

for all T\scriptstyle\hat\Sigma is a measure on \scriptstyle\hat\Sigma known as the completion of μ.

References[edit]

  • 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)