# Outer measure

Outer measure (Redirected from Carathéodory's theorem (measure theory)) Jump to navigation Jump to search This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (January 2013) (Learn how and when to remove this template message) In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures.[1][2] Carathéodory's work on outer measures found many applications in measure-theoretic set theory (outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory.

Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than intervals in {displaystyle mathbb {R} } or balls in {displaystyle mathbb {R} ^{3}} . One might expect to define a generalized measuring function {displaystyle varphi } on {displaystyle mathbb {R} } that fulfills the following requirements: Any interval of reals [a, b] has measure b − a The measuring function {displaystyle varphi } is a non-negative extended real-valued function defined for all subsets of {displaystyle mathbb {R} } . Translation invariance: For any set A and any real x, the sets A and A+x have the same measure (where {displaystyle A+x={a+x:ain A}} ) Countable additivity: for any sequence (Aj) of pairwise disjoint subsets of {displaystyle mathbb {R} } {displaystyle varphi left(bigcup _{i=1}^{infty }A_{i}right)=sum _{i=1}^{infty }varphi (A_{i}).} It turns out that these requirements are incompatible conditions; see non-measurable set. The purpose of constructing an outer measure on all subsets of X is to pick out a class of subsets (to be called measurable) in such a way as to satisfy the countable additivity property.

Contents 1 Outer measures 2 Measurability of sets relative to an outer measure 2.1 The measure space associated to an outer measure 3 Restriction and pushforward of an outer measure 3.1 Pushforward 3.2 Restriction 3.3 Measurability of sets relative to a pushforward or restriction 4 Regular outer measures 4.1 Definition of a regular outer measure 4.2 The regular outer measure associated to an outer measure 5 Outer measure and topology 6 Construction of outer measures 6.1 Method I 6.2 Method II 7 See also 8 Notes 9 References 10 External links Outer measures Given a set {displaystyle X,} let {displaystyle 2^{X}} denote the collection of all subsets of {displaystyle X,} including the empty set {displaystyle varnothing .} An outer measure on {displaystyle X} is a set function {displaystyle mu :2^{X}to [0,infty ]} such that null empty set: {displaystyle mu (varnothing )=0} countably subadditive: for arbitrary subsets {displaystyle A,B_{1},B_{2},ldots } of {displaystyle X,} {displaystyle {text{if }}Asubseteq bigcup _{j=1}^{infty }B_{j}{text{ then }}mu (A)leq sum _{j=1}^{infty }mu (B_{j}).} Note that there is no subtlety about infinite summation in this definition. Since the summands are all assumed to be nonnegative, the sequence of partial sums could only diverge by increasing without bound. So the infinite sum appearing in the definition will always be a well-defined element of {displaystyle [0,infty ].} If, instead, an outer measure were allowed to take negative values, its definition would have to be modified to take into account the possibility of non-convergent infinite sums.

An alternative and equivalent definition.[3] Some textbooks, such as Halmos (1950), instead define an outer measure on {displaystyle X} to be a function {displaystyle mu :2^{X}to [0,infty ]} such that null empty set: {displaystyle mu (varnothing )=0} monotone: if {displaystyle A} and {displaystyle B} are subsets of {displaystyle X} with {displaystyle Asubseteq B,} then {displaystyle mu (A)leq mu (B)} for arbitrary subsets {displaystyle B_{1},B_{2},ldots } of {displaystyle X,} {displaystyle mu left(bigcup _{j=1}^{infty }B_{j}right)leq sum _{j=1}^{infty }mu (B_{j}).} show Proof of equivalence. Measurability of sets relative to an outer measure Let {displaystyle X} be a set with an outer measure {displaystyle mu .} One says that a subset {displaystyle E} of {displaystyle X} is {displaystyle mu } -measurable (sometimes called Carathéodory-measurable relative to {displaystyle mu ,} after the mathematician Carathéodory) if and only if {displaystyle mu (A)=mu (Acap E)+mu (Asetminus E)} for every subset {displaystyle A} of {displaystyle X.} Informally, this says that a {displaystyle mu } -measurable subset is one which may be used as a building block, breaking any other subset apart into pieces (namely, the piece which is inside of the measurable set together with the piece which is outside of the measurable set). In terms of the motivation for measure theory, one would expect that area, for example, should be an outer measure on the plane. One might then expect that every subset of the plane would be deemed "measurable," following the expected principle that {displaystyle operatorname {area} (Acup B)=operatorname {area} (A)+operatorname {area} (B)} whenever {displaystyle A} and {displaystyle B} are disjoint subsets of the plane. However, the formal logical development of the theory shows that the situation is more complicated. A formal implication of the axiom of choice is that for any definition of area as an outer measure which includes as a special case the standard formula for the area of a rectangle, there must be subsets of the plane which fail to be measurable. In particular, the above "expected principle" is false, provided that one accepts the axiom of choice.

The measure space associated to an outer measure It is straightforward to use the above definition of {displaystyle mu } -measurability to see that if {displaystyle Asubseteq X} is {displaystyle mu } -measurable then its complement {displaystyle Xsetminus Asubseteq X} is also {displaystyle mu } -measurable.

The following condition is known as the "countable additivity of {displaystyle mu } on measurable subsets."

if {displaystyle A_{1},A_{2},ldots } are {displaystyle mu } -measurable subsets of {displaystyle X} and {displaystyle A_{i}cap A_{j}} is empty whenever {displaystyle ineq j,} then one has {displaystyle mu {Big (}bigcup _{j=1}^{infty }A_{j}{Big )}=sum _{j=1}^{infty }mu (A_{j}).} show Proof of countable additivity.

A similar proof shows that: if {displaystyle A_{1},A_{2},ldots } are {displaystyle mu } -measurable subsets of {displaystyle X,} then the union {displaystyle A_{1}cup A_{2}cup cdots } and intersection {displaystyle A_{1}cap A_{2}cap cdots } are also {displaystyle mu } -measurable.

The properties given here can be summarized by the following terminology: Given any outer measure {displaystyle mu } on a set {displaystyle X,} the collection of all {displaystyle mu } -measurable subsets of {displaystyle X} is a σ-algebra. The restriction of {displaystyle mu } to this {displaystyle sigma } -algebra is a measure.

One thus has a measure space structure on {displaystyle X,} arising naturally from the specification of an outer measure on {displaystyle X.} This measure space has the additional property of completeness, which is contained in the following statement: Every subset {displaystyle Asubseteq X} such that {displaystyle mu (A)=0} is {displaystyle mu } -measurable.

This is easy to prove by using the second property in the "alternative definition" of outer measure.

Restriction and pushforward of an outer measure Let μ be an outer measure on the set X.

Pushforward Given another set Y and a map f : X→Y, define f# μ : 2Y→[0,∞] by {displaystyle {big (}f_{sharp }mu {big )}(A)=mu {big (}f^{-1}(A){big )}.} One can verify directly from the definitions that f# μ is an outer measure on Y.

Restriction Let B be a subset of X. Define μB : 2X→[0,∞] by {displaystyle mu _{B}(A)=mu (Acap B).} One can check directly from the definitions that μB is another outer measure on X.

Measurability of sets relative to a pushforward or restriction If a subset A of X is μ-measurable, then it is also μB-measurable for any subset B of X.

Given a map f : X→Y and a subset A of Y, if f −1(A) is μ-measurable then A is f# μ-measurable. More generally, f −1(A) is μ-measurable if and only if A is f# (μB)-measurable for every subset B of X.

Regular outer measures Definition of a regular outer measure Given a set X, an outer measure μ on X is said to be regular if any subset can be approximated 'from the outside' by μ-measurable sets. Formally, this is requiring either of the following equivalent conditions: for any subset A of X and any positive number ε, there exists a μ-measurable subset B of X which contains A and with μ(B) < μ(A) + ε. for any subset A of X, there exists a μ-measurable subset B of X which contains A and such that μ(B) = μ(A). It is automatic that the second condition implies the first; the first implies the second by considering the intersection of a minimizing sequence of subsets. The regular outer measure associated to an outer measure Given an outer measure μ on a set X, define ν : 2X→[0,∞] by {displaystyle nu (A)=inf {Big {}mu (B):mu {text{-measurable subsets }}Bsubset X{text{ with }}Bsupset A{Big }}.} Then ν is a regular outer measure on X which assigns the same measure as μ to all μ-measurable subsets of X. Every μ-measurable subset is also ν-measurable, and every ν-measurable subset of finite ν-measure is also μ-measurable. So the measure space associated to ν may have a larger σ-algebra than the measure space associated to μ. The restrictions of ν and μ to the smaller σ-algebra are identical. The elements of the larger σ-algebra which are not contained in the smaller σ-algebra have infinite ν-measure and finite μ-measure. From this perspective, ν may be regarded as an extension of μ. Outer measure and topology Suppose (X, d) is a metric space and φ an outer measure on X. If φ has the property that {displaystyle varphi (Ecup F)=varphi (E)+varphi (F)} whenever {displaystyle d(E,F)=inf{d(x,y):xin E,yin F}>0,} then φ is called a metric outer measure.

Theorem. If φ is a metric outer measure on X, then every Borel subset of X is φ-measurable. (The Borel sets of X are the elements of the smallest σ-algebra generated by the open sets.) Construction of outer measures See also: Valuation (measure theory) There are several procedures for constructing outer measures on a set. The classic Munroe reference below describes two particularly useful ones which are referred to as Method I and Method II.

Method I Let X be a set, C a family of subsets of X which contains the empty set and p a non-negative extended real valued function on C which vanishes on the empty set.

Theorem. Suppose the family C and the function p are as above and define {displaystyle varphi (E)=inf {biggl {}sum _{i=0}^{infty }p(A_{i}),{bigg |},Esubseteq bigcup _{i=0}^{infty }A_{i},forall iin mathbb {N} ,A_{i}in C{biggr }}.} That is, the infimum extends over all sequences {Ai} of elements of C which cover E, with the convention that the infimum is infinite if no such sequence exists. Then φ is an outer measure on X.

Method II The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures. Suppose (X, d) is a metric space. As above C is a family of subsets of X which contains the empty set and p a non-negative extended real valued function on C which vanishes on the empty set. For each δ > 0, let {displaystyle C_{delta }={Ain C:operatorname {diam} (A)leq delta }} and {displaystyle varphi _{delta }(E)=inf {biggl {}sum _{i=0}^{infty }p(A_{i}),{bigg |},Esubseteq bigcup _{i=0}^{infty }A_{i},forall iin mathbb {N} ,A_{i}in C_{delta }{biggr }}.} Obviously, φδ ≥ φδ' when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus {displaystyle lim _{delta rightarrow 0}varphi _{delta }(E)=varphi _{0}(E)in [0,infty ]} exists (possibly infinite).

Theorem. φ0 is a metric outer measure on X.

This is the construction used in the definition of Hausdorff measures for a metric space.

See also Inner measure Notes ^ Carathéodory 1968 ^ Aliprantis & Border 2006, pp. S379 ^ The original definition given above follows the widely cited texts of Federer and of Evans and Gariepy. Note that both of these books use non-standard terminology in defining a "measure" to be what is here called an "outer measure." References Aliprantis, C.D.; Border, K.C. (2006). Infinite Dimensional Analysis (3rd ed.). Berlin, Heidelberg, New York: Springer Verlag. ISBN 3-540-29586-0. Carathéodory, C. (1968) [1918]. Vorlesungen über reelle Funktionen (in German) (3rd ed.). Chelsea Publishing. ISBN 978-0828400381. Evans, Lawrence C.; Gariepy, Ronald F. (2015). Measure theory and fine properties of functions. Revised edition. Textbooks in Mathematics. CRC Press, Boca Raton, FL. pp. xiv+299. ISBN 978-1-4822-4238-6. Federer, H. (1996) [1969]. Geometric Measure Theory. Classics in Mathematics (1st ed reprint ed.). Berlin, Heidelberg, New York: Springer Verlag. ISBN 978-3540606567. Halmos, P. (1978) [1950]. Measure theory. Graduate Texts in Mathematics (2nd ed.). Berlin, Heidelberg, New York: Springer Verlag. ISBN 978-0387900889. Munroe, M. E. (1953). Introduction to Measure and Integration (1st ed.). Addison Wesley. ISBN 978-1124042978. Kolmogorov, A. N.; Fomin, S. V. (1970). Introductory Real Analysis. Richard A. Silverman transl. New York: Dover Publications. ISBN 0-486-61226-0. External links Outer measure at Encyclopedia of Mathematics Caratheodory measure at Encyclopedia of Mathematics show vte Measure theory Categories: Measures (measure theory)

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