Borel measure

Without the condition of regularity the Borel measure need not be unique.

It is also the completion of the Borel measure, as in the one-dimensional case.

Assuming compactness one may extend it to a regular Borel measure.

It is the algebra on which the Borel measure is defined.

The first chapter develops the theory of measure (see Borel measure).

The Borel measure is translation-invariant, but not complete.

In practice Baire measures can be replaced by regular Borel measures.

Any measure defined on the Borel sets is called a Borel measure.

First, the decomposition of the singular part of a regular Borel measure on the real line can be refined:

The relation between Baire measures and regular Borel measures is as follows:

Specifically, let X be a locally compact Hausdorff space equipped with a positive Borel measure.

The real line with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it.

A finite Baire measure on a compact space is the restriction of a unique regular Borel measure.

Let D denote the unit disc in the complex plane C, equipped with some Borel measure μ.

In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on [0, ).

The resulting natural positive Borel measure allows one to develop a theory of integrating functions on the manifold by means of the associated Lebesgue integral.

Malliavin's lemma gives a sufficient condition for a finite Borel measure to be absolutely continuous with respect to Lebesgue measure.

Let be a metric measure space, where d is a metric on X and μ is a Borel measure.

The union of k or fewer null sets in an atomless σ-finite Borel measure on a Polish space is null.

Lebesgue measure is an example of a complete measure; in some constructions, it's defined as the completion of a non-complete Borel measure.

This definition makes sense if x is an integrable function (in L), a compactly supported distribution, or is a finite Borel measure.

For every compact Hausdorff space, every finite Baire measure has a unique extension to a regular Borel measure.

The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure.

It is known, for instance, that every continuous translation invariant continuous linear operator on L is the convolution with a finite Borel measure.

The volume form also gives a way to integrate functions on the manifold, and this geometric integral agrees with the integral obtained by the canonical Borel measure.