nLab
Borel measure

Contents

Contents

Idea

Borel measures are measures on the Borel σ-algebra? of a topological space. They play an important role in measure theory.

Definition

A Borel measure on XX is a (countably additive) measure on the Borel σ-algebra? of the topological space XX.

A Baire measure on XX is a (countably additive) measure on the Baire σ-algebra? of the topological space XX.

A Radon measure on XX is a Borel measure μ\mu on XX such that for any Borel subset BXB\subset X and any ϵ>0\epsilon\gt0 there is a compact subset KBK\subset B such that |μ|(BK)<ϵ|\mu|(B\setminus K)\lt\epsilon, where |μ||\mu| denotes the total variation? of the measure μ\mu.

Borel measures are uniquely determined by their values on open subsets, and are closely related to continuous valuations on the underlying locale.

A measure μ\mu on a topological space XX is tight if for any ϵ>0\epsilon\gt0 there is a compact subset KXK\subset X such that |μ|(A)<ϵ|\mu|(A)\lt\epsilon for any AXKA\subset X\setminus K.

A measure μ\mu on a topological space XX is regular if for any measurable subset AXA\subset X and ϵ>0\epsilon\gt0 there is a closed subset FAF\subset A such that AFA\setminus F is measurable and μ(AF)<ϵ\mu(A\setminus F)\lt\epsilon.

Radon measures on Hausdorff spaces are regular and tight. Regular tight Borel measures are automatically Radon. A regular Borel measure need not be tight.

A Borel measure μ\mu on a topological space XX is τ-additive (alias τ-regular, τ-smooth) if |μ|( iU i)=lim i|μ|(U i)|\mu|(\bigcup_i U_i)=\lim_i |\mu|(U_i) for any directed system of open subsets U iXU_i\subset X. In other words, the underlying valuation of μ\mu is a continuous valuation.

If the above condition only holds in the case iU i=X\bigcup_i U_i=X, we talk about τ 0\tau_0-additive (or weakly τ-additive) measures.

Any regular τ 0\tau_0-additive Borel measure is τ-additive. There are τ 0\tau_0-additive Borel measures that are not τ-additive.

Properties

On a metric space every Borel measure is regular. More generally, every Borel measure on a perfectly normal? space is regular.

On a complete separable metric space every Borel measure is Radon.

Every Baire measure is regular.

Every Radon measure is τ-additive.

Every τ-additive measure on a regular space is regular. In particular, every τ-additive measure on a compact space is Radon.

Every tight τ-additive measure is Radon.

Every Borel measure on a separable metric space X is τ-additive. Moreover, this is true if X is hereditary Lindelöf?.

Every τ-additive measure has a support, which is a closed subset.

Extension properties

Every tight Baire measure on a completely regular space admits a unique extension to a Radon measure. More generally, any tight Baire measure on a Hausdorff space has a Radon extension.

Every Baire measure on a σ-compact completely regular space has a unique extension to a Radon measure.

Relation to linear functionals

Given a topological space XX, the integration map sends Borel measures on XX to linear functionals on the space of continuous bounded functions on XX.

For many types of topological spaces and measures this map is bijective, as indicated below.

There is a bijection between Baire measures on a topological space and continuous linear functionals LL on continuous bounded functions such that L(f n)0L(f_n)\to 0 whenever f n0f_n \to 0 pointwise and monotonic.

There is a bijection between Radon measures on a completely regular topological space and continuous linear functionals LL on continuous bounded functions such that for any ϵ>0\epsilon\gt0 there is a compact subset KK such that for any continuous bounded function ff that vanishes on KK we have |L(f)|ϵsup|f||L(f)|\le\epsilon\sup|f|.

There is a bijection between τ-additive Borel measures on a completely regular topological space and continuous linear functionals LL on continuous bounded functions such that L(f a)0L(f_a)\to0 for any net ff of bounded continuous functions that decreases to zero pointwise.

There is a bijection between positive Borel measures on a locally compact topological space that are inner regular on all Borel subsets with respect to compact subsets and positive linear functionals on continuous functions vanishing at infinity.

Last revised on June 27, 2019 at 16:41:40. See the history of this page for a list of all contributions to it.