Outer measure Guide, Meaning , Facts, Information and Description

In mathematics, in particular in measure theory, an outer measure is a function defined on all subsets of a given set with values in the extended real numbers. Outer measures are used to define measurable sets and countably additive measures. 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. The initial idea is 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. 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 non-measurable set. The purpose of constructing an outer measure is to define which sets are measurable, and fulfil the countably additivity axiom.

Formally, an outer measure is defined as a function defined on all subsets of a set X

such that

• The empty set has zero outer measure (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 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 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 Lebesgue measure that is so important for measure theory and the theory of integrals.

Table of contents

1 Outer measure and topology 2 Construction of outer measures 3 References

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 Borel setss 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 emptyset and p an extended real valued function on C which vanishes on the emptyset.

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 emptyset and p an extended real valued function on C which vanishes on the emptyset. 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 Hausdorff measures for a metric space.

References

• P. Halmos, Measure theory, D. van Nostrand and Co., 1950
• M. E. Munroe, Introduction to Measure and Integration, Addison Wesley, 1953

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