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

where

Equivalently, one can simply say that

We give three equivalent definitions of Radon measures.

Equivalence of definitions

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

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

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

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.

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.,

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

(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

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.

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