Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## 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 $M$.

###### 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 $X$.

## Examples

Most measures of interest in geometry are Radon. For example