outer measure

There are several procedures for constructing outer measures on a set.

The outer measure of is equal to 1, but the inner measure may differ.

Then φ is an outer measure on X.

Due to the metric outer measure property, all Borel subsets of are measurable.

Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra.

Outer measures are not, in general, measures, since they may fail to be σ-additive.)

-finite measures, and measures arising as the restriction of outer measures, are saturated.

The Lebesgue outer measure on R is an example of a Borel regular measure.

Its statement is as follows: Let denote the Lebesgue outer measure on , and let .

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

Outer measure: the proof of Carathéodory's extension theorem is based upon the outer measure concept.

In mathematics, an outer measure μ on n-dimensional Euclidean space R is called Borel regular if the following two conditions hold:

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

It can be seen that is an outer measure (more precisely, it is a metric outer measure).

The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures.

A general theory of outer measures was first introduced by Constantin Carathéodory to provide a basis for the theory of measurable sets and countably additive measures.

The measure-category duality provides a measure analogue of Luzin sets - sets of positive measure, every uncountable subset of which has positive outer measure.

Here the "objects" should be sets of finite measure (or, in fact, just of finite outer measure) for the notion of "dividing the volume in half" to make sense.

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.

In mathematics, in particular in 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.

It can be proved that a Borel Regular measure, although introduced here as an outer measure (only countably subadditive), becomes a full measure (countably additive) if restricted to the Borel sets.

An outer measure satisfying only the first of these two requirements is called a Borel measure, while an outer measure satisfying only the second requirement is called a regular measure.

In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0, ] to each set in R or, more generally, in any metric space.

The first principle is based on the fact that the inner measure and outer measure are equal for measurable sets, the second is based on Lusin's theorem, and the third is based on Egorov's theorem.

A Bernstein set partitions the real line into two pieces in a peculiar way: every set of positive outer measure meets both the Bernstein set and its complement, as does every set with the property of Baire that is not a meagre set.