nLab Radon measure

Definition

NB: In the following the symbols “$\subset$” and “$\supset$” are used to denote nonstrict inclusions of subsets, sometimes also denoted by “$\subseteq$” and “$\supseteq$”.

Recall the following properties of a Borel measure $\mu$ on a Hausdorff topological space:

• $\mu$ is outer regular if for every Borel subset $B$ we have

  $$\mu(B)=\inf\{\mu(V)\mid V\supset B\; and\; V\; is\; open\}.$$

• $\mu$ is locally finite if every point has a neighborhood with a finite $\mu$-measure.

• $\mu$ is inner regular on some Borel subset $B$ if

  $$\mu(B)=\sup\{\mu(K)\mid K\subset B\; and\; K\; is\; compact\}.$$

Also, if $m$ and $M$ are Borel measures, then $m$ is the essential measure associated with $M$ if

$$m(A)=\sup\{m^*(C)\mid C\subset A,\; m^*(C)\; is\; finite\},$$

where

$$m^*(C)=\inf\{m(B)\mid B\supset C\; and\; B\; is\; Borel\}.$$

Equivalently, one can simply say that

$$m(B)=\sup\{M(B')\mid B'\subset B,\; M(B')\; is\; finite,\; B'\; is\; Borel\}.$$

We give three equivalent definitions of Radon measures.

Definition

If $X$ is a Hausdorff topological space, then a Radon measure on $X$ is a Borel measure $m$ on $X$ such that $m$ is locally finite and inner regular on all Borel subsets.

Definition

If $X$ is a Hausdorff topological space, then a Radon measure on $X$ is a Borel measure $M$ on $X$ such that $M$ is locally finite, outer regular, and inner regular on all open subsets.

Definition

If $X$ is a Hausdorff topological space, then a Radon measure on $X$ is a pair of Borel measures $m$ and $M$ on $X$ such that $m$ is the essential measure associated with $M$, $M$ is outer regular (on all Borel subsets), $M$ is locally finite, $M$ is inner regular on all Borel subsets, and $m(B)=M(B)$ whenever $B$ is open or $M(B)$ is finite.

Equivalence of definitions

In order to pass from $m$ to $M$, set

$$M(B)=\inf\{m(V)\mid V\supset B\; and\; V\; is\; open\}.$$

In order to pass from $M$ to $m$, set

$$m(B)=\sup\{M(B')\mid B'\subset B,\; M(B')\; is\; finite,\; B'\; is\; Borel\}.$$

If $m(X)$ or $M(X)$ is finite, then $m=M$.

If $B$ is a Borel subset such that $M(B)$ is finite or $B$ is open, then $m(B)=M(B)$.

A Radon measure is σ-finite if $m$ is σ-finite.

A Radon measure is moderated if $M$ is σ-finite.

Real and complex Radon measures

Definition

A real (respectively complex) Radon measure on a Hausdorff topological space $X$ is a real (respectively complex) valued function $\mu$ defined on relatively compact Borel subsets of $X$ that is (1) countably additive, (2) every relative compact Borel subset $B\subset X$ can be presented as the union of countably many compact subsets and a subset $N\subset B$ such that $\mu(N')=0$ for any Borel subset $N'\subset N$, and (3) any point has a neighborhood $V$ such that $\sup |\mu(B)|$ is finite, where $B\subset V$ is a relatively compact subset of $X$.

If $\mu\ge0$, then $\mu$ can be extended to a Radon measure $(m,M)$ in the previous sense.

On locally compact Hausdorff topological spaces

Radon measures on locally compact Hausdorff topological spaces admit yet another, Daniell-style definition, which is explored in detail in Bourbaki’s book.

Definition

Suppose $X$ is a locally compact Hausdorff topological space. A Radon measure on $X$ is a positive linear functional

$$\mu\colon C_c(X,\mathbf{R})\to\mathbf{R},$$

where $C_c$ refers to the vector space of continuous compactly supported functions.

Such $\mu$ induces a pair $(m,M)$ in the sense of above definitions as follows: $M=\mu^*$ and $m=\mu^{\bar*}$, where $\mu^*$ is the outer measure associated to $\mu$, i.e.,

$$\mu^*(A)=\inf\{\mu(\chi_B)\mid B\supset A,\; B\; is\; Borel\}$$

and $\mu^{\bar*}$ is the essential measure associated to $\mu^*$:

$$\mu^{\bar*}(A)=\sup\{\mu^*(C)\mid C\subset A,\; \mu^*(C)\; is\; finite\}.$$

(For σ-finite spaces we have $\mu^*=\mu^{\bar*}$.)

Vice versa, given $M$, we can reconstruct $\mu$ as the integral with respect to $M$.

Pushforwards of Radon measures

Definition

Suppose $X$ and $Y$ are Hausdorff topological spaces and $\mu=(m,M)$ is a Radon measure on $X$. A map of sets $H\colon X\to Y$ is Lusin $\mu$-measurable if for any compact $K\subset X$ and $\epsilon\gt0$ there is a compact $K'\subset K$ such that $\mu(K\setminus K')\lt\epsilon$ and the restriction of $H$ to $K'$ is continuous. We say that $H$ is $\mu$-proper if, in addition, every point of $Y$ has a neighborhood whose inverse image is $\mu$-integrable.

For example, continuous maps are Lusin $\mu$-measurable for any $\mu$.

Lusin $\mu$-measurable maps form a sheaf with respect to $X$.

Any Lusin $\mu$-measurable map is Borel $\mu$-measurable (meaning preimages of Borel sets are $\mu$-measurable sets).

If $Y$ is metrizable and separable, then any Borel $\mu$-measurable map is Lusin $\mu$-measurable.

Step functions and lower semicontinuous maps are always Lusin $\mu$-measurable.

Definition

Suppose $H\colon X\to Y$ is a $\mu$-proper map. The pushforward measure $H_*\mu$ is a Radon measure on $Y$ defined as follows. Given a Borel subset $B\subset Y$, we set

$$H_*\mu(B)=\mu^\bullet(H^*(B)).$$

This yields a Radon measure $m$ on $Y$.

Theorem

If $H\colon X\to Y$ is a continuous map of Hausdorff topological spaces, then a Radon measure $\nu$ on $Y$ is the pushforward along $H$ of a Radon measure on $X$ if and only if $\nu$ is concentrated in a countable union of images under $H$ of compact subsets of $X$.

Properties

• If a Radon measure is $\sigma$-finite then it is regular (i.e. both inner and outer regular) on all Borel subsets.

• A Radon measure on a Hausdorff space is τ-additive. The converse is true on a compact Hausdorff space.

• Radon probability measures on compact Hausdorff spaces form a monad: the Radon monad. Just as well, Radon probability measures of finite first moment on complete metric spaces give the Kantorovich monad.

(See also monads of probability, measures and valuations.)

Examples

Most measures of interest in geometry are Radon. For example

• The Dirac measures.

• The Lebesgue measure on the real line.

• The measure associated to a volume form on a Riemannian manifold.

• The left (right) Haar measure on a locally compact topological group is a nonzero Radon measure which is invariant under left (right) multiplications by elements in the group.

See also

• Radon monad, Kantorovich monad, monads of probability, measures, and valuations

• Borel measure, τ-additive measure, continuous valuation

References

The canonical references on Radon measures are

• Laurent Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures.

• Nicolas Bourbaki, Integration. Chapter IX.

More recent expositions include

• V. Bogachev, Measure Theory, vol. 2 (2007). doi:10.1007/978-3-540-34514-5

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995