In Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. One reason is that... mathematics, in particular in In mathematics, a measure is a function that assigns a number, e.g., a size, volume, or probability, to subsets of a given set. The concept is important in mathematical analysis and probability theory. Measure theory is that branch of real analysis which investigates sigma algebras, measures, measurable functions and... measure theory, an outer measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. A general theory of outer measures was developed by Carathéodory to provide a basis for the theory of In mathematics, a measure is a function that assigns a number, e.g., a size, volume, or probability, to subsets of a given set. The concept is important in mathematical analysis and probability theory. Measure theory is that branch of real analysis which investigates sigma algebras, measures, measurable functions and... measurable sets and countably additive measures. Carathéodory's work on outer measures found many applications in measure-theoretic set theory and was used in an essential way by Felix Hausdorff (November 8, 1868 - January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory and functional analysis. He defined and studied partially ordered sets, Hausdorff spaces, and the Hausdorff dimension. He proved the... Hausdorff to define a dimension-like metric invariant now called In mathematics, the Hausdorff dimension is an extended non-negative real number, that is in the closed infinite interval [0, ∞], associated to any metric space . It was introduced in 1918 by the mathematician Felix Hausdorff. Many of the technical developments used to compute the Hausdorff dimension for highly irregular... Hausdorff dimension.

Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than mere intervals or open balls in R³. One might expect to define a generalized measuring function φ that fulfils the following three requirements:

1. Any interval of reals [a, b] has measure b − a
2. The measuring function φ is a non-negative extended real-valued function defined for all subsets of R.
3. In mathematics, a measure is a function that assigns a number, e.g., a size, volume, or probability, to subsets of a given set. The concept is important in mathematical analysis and probability theory. Measure theory is that branch of real analysis which investigates sigma algebras, measures, measurable functions and... Countable additivity, For any sequence {A_j}_j of pairwise disjoint subsets of X

It turns out the second and third requirements together for all sets are incompatible conditions; see In mathematics, a non-measurable set is a set whose structure is so complicated it sheds light on the very notion of length, area or volume. This page will give a non-technical description of this concept. For a technical description see measure (mathematics) and the various constructions of non... non-measurable set. The purpose of constructing an outer measure on all subsets of X is to suitably pick out a class of subsets (to be called measurable) in such a way that fulfils the countably additivity property.

Formal definitions

An outer measure is defined as a function defined on all subsets of a set X

such that

• The In mathematics, the empty set is the set with no elements. Notation The standard notation for denoting the empty set, invented by Nicholas Bourbaki, is the symbol , also written as or ∅, and sometimes approximated by the glyph Ø, (not to be confused with the Greek letter φ). However, for wider... empty set has zero outer measure ( Let μ be a measure on a sigma algebra Σ of subsets of a set X. An element A in Σ is said to have measure zero if μ(A)=0. Any set of measure zero is a null set. The opposite is not true, because a null set is... measure zero).

• Monotonicity

• Countable sub-additivity: for any sequence {A_j}_j of subsets of X (pairwise disjoint or not)

This allows us to define the concept of measurability as follows: a subset E of X is φ-measurable (or Carathéodory-measurable by φ) In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. Although P iff Q is most standard, common alternative phrases include P is necessary and sufficient for Q and P... iff for every subset A of X

Theorem. The φ-measurable sets form a σ-algebra and φ restricted to the measurable sets is a countably additive In measure theory, a complete measure is a measure in which every subset of every null set is measurable (having measure 0). Every measure has an extension that is complete. The smallest such extension is called the completion of the measure. Suppose μ is a measure on some set X... complete measure.

For a proof of this theorem see the Halmos reference, section 11.

This method is known as the Carathéodory construction and is one way of arriving at the concept of Henri Léon Lebesgue (June 28, 1875 - July 26, 1941) was a French mathematician, most famous for his theory of integration. Lebesgues integration theory was originally published in his dissertation, Intégral, longueur, aire (Integral, length, area), at the University of Nancy in 1902. Lebesgues father was a... Lebesgue measure that is so important for Measure can mean: To perform a measurement. In mathematics, a measure is a way to assign non-negative real numbers to subsets of a given set, in order to measure their sizes or probabilities. See measure (mathematics) for a treatment of the concept. In music, a measure is a unit... measure theory and the theory of This article deals with the concept of an integral in calculus. For other meanings of integral see integration. In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. Unlike the process of differentiation, there are several different definitions of integration, all of which... integrals.

Outer measure and topology

Suppose (X, d) is a metric space and φ an outer measure on X. If φ has the property that

whenever

then φ is called a metric outer measure. The In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. The minimal σ-algebra containing the compact sets. The minimal σ-algebra on a set X containing a... Borel sets of X are the elements of the smallest σ-algebra generated by the open sets.

Theorem. If φ is a metric outer measure on X, then every Borel subset of X is φ-measurable.

Construction of outer measures

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.

Let X be a set, C a subset of 2^X which contains the empty set and p an extended real valued function on C which vanishes on the empty set.

Theorem. Suppose the class C and the function p are as above and define

where the infimum extends over all sequences {A_i}_i of elements of C which cover E (with the convention that if no such sequence exists, then the infimum is infinite). Then φ is an outer measure on X.

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 subset of 2^X which contains the empty set and p an extended real valued function on C which vanishes on the empty set. For each δ > 0, let

and

where the infimum extends over all sequences {A_i}_i of elements of C_δ which cover E. Obviously, φ_δ ≥ φ_δ' when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus

exists.

Theorem. φ₀ is a metric outer measure on X.

This is the construction used in the definition of In mathematics, the Hausdorff dimension is an extended non-negative real number, that is in the closed infinite interval [0, ∞], associated to any metric space . It was introduced in 1918 by the mathematician Felix Hausdorff. Many of the technical developments used to compute the Hausdorff dimension for highly irregular... Hausdorff measures for a metric space.

References

• Paul Halmos Paul Richard Halmos (born March 3, 1916) is a Hungarian-born American mathematician who has done research in the fields of probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular Hilbert spaces). He is noted for a number of expository books, viewed by many to... P. Halmos, Measure theory, D. van Nostrand and Co., 1950
• M. E. Munroe, Introduction to Measure and Integration, Addison Wesley, 1953