nLab Borel measure

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

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

A Radon measure on $X$ is a Borel measure $\mu$ on $X$ such that for any Borel subset $B\subset X$ and any $\epsilon\gt0$ there is a compact subset $K\subset B$ such that $|\mu|(B\setminus K)\lt\epsilon$, where $|\mu|$ denotes the total variation? of the measure $\mu$.

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

A measure $\mu$ on a topological space $X$ is tight if for any $\epsilon\gt0$ there is a compact subset $K\subset X$ such that $|\mu|(A)\lt\epsilon$ for any $A\subset X\setminus K$.

A measure $\mu$ on a topological space $X$ is regular if for any measurable subset $A\subset X$ and $\epsilon\gt0$ there is a closed subset $F\subset A$ such that $A\setminus F$ is measurable and $\mu(A\setminus F)\lt\epsilon$.

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 $\mu$ on a topological space $X$ is τ-additive (alias τ-regular, τ-smooth) if $|\mu|(\bigcup_i U_i)=\lim_i |\mu|(U_i)$ for any directed system of open subsets $U_i\subset X$. In other words, the underlying valuation of $\mu$ is a continuous valuation.

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

Any regular $\tau_0$-additive Borel measure is τ-additive. There are $\tau_0$-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 $X$, the integration map sends Borel measures on $X$ to linear functionals on the space of continuous bounded functions on $X$.

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 $L$ on continuous bounded functions such that $L(f_n)\to 0$ whenever $f_n \to 0$ pointwise and monotonic.

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

There is a bijection between τ-additive Borel measures on a completely regular topological space and continuous linear functionals $L$ on continuous bounded functions such that $L(f_a)\to0$ for any net $f$ 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.

Related concepts

• Borel set

• Radon measure

• continuous valuation

• measure