Inner measure

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.

Contents

• 
  1 Definition
  


• 
  2 The inner measure induced by a measure
  


• 
  3 Measure completion
  


• 
  4 References
  

Definition

An inner measure is a function

φ:2X→[0,∞],{displaystyle varphi :2^{X}rightarrow [0,infty ],}

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

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

φ(∅)=0{displaystyle varphi (varnothing )=0}

• 
  Superadditive: For any disjoint sets A and B,

φ(A∪B)≥φ(A)+φ(B).{displaystyle varphi (Acup B)geq varphi (A)+varphi (B).}

• Limits of decreasing towers: For any sequence {A_j} of sets such that Aj⊇Aj+1{displaystyle A_{j}supseteq A_{j+1}} for each j and φ(A1)<∞{displaystyle varphi (A_{1})<infty }
  

φ(⋂j=1∞Aj)=limj→∞φ(Aj){displaystyle varphi left(bigcap _{j=1}^{infty }A_{j}right)=lim _{jto infty }varphi (A_{j})}

• Infinity must be approached: If φ(A)=∞{displaystyle 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≤φ(B)<∞{displaystyle cleq varphi (B)<infty }

The inner measure induced by a measure

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

μ∗(T)=sup{μ(S):S∈Σ and S⊆T}.{displaystyle mu _{*}(T)=sup{mu (S):Sin Sigma {text{ and }}Ssubseteq 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 E ⊆ F then μ_*(E) ≤ μ_*(F).

Measure completion

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 finite measure defined on a σ-algebra Σ over X and μ^* and μ_* are corresponding induced outer and inner measures, then the sets T ∈ 2^X such that μ_*(T) = μ^* (T) form a σ-algebra Σ^{displaystyle {hat {Sigma }}} with Σ⊆Σ^{displaystyle Sigma subseteq {hat {Sigma }}}.^[1] The set function μ̂ defined by

μ^(T)=μ∗(T)=μ∗(T){displaystyle {hat {mu }}(T)=mu ^{*}(T)=mu _{*}(T)},

for all T∈Σ^{displaystyle Tin {hat {Sigma }}} is a measure on Σ^{displaystyle {hat {Sigma }}} known as the completion of μ.

References

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, .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  ISBN 0-486-61226-0 (Chapter 7)

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