nLab
Radon measure

Contents

Contents

Definition

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

  • μ\mu is outer regular if for every Borel subset BB we have

    μ(B)=inf{μ(V)VBandVisopen}.\mu(B)=\inf\{\mu(V)\mid V\supset B\; and\; V\; is\; open\}.
  • μ\mu is locally finite if every point has a neighborhood with a finite μ\mu-measure.

  • μ\mu is inner regular on some Borel subset BB if

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

Also, if mm and MM are Borel measures, then mm is the essential measure associated with MM if

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

where

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

Equivalently, one can simply say that

m(B)=sup{M(B)BB,M(B)isfinite,BisBorel}. 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 XX is a Hausdorff topological space, then a Radon measure on XX is a Borel measure mm on XX such that mm is locally finite and inner regular on all Borel subsets.

Definition

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

Definition

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

Equivalence of definitions

In order to pass from mm to MM, set

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

In order to pass from MM to mm, set

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

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

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

A Radon measure is σ-finite if mm is σ-finite.

A Radon measure is moderated if MM is σ-finite.

Real and complex Radon measures

Definition

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

If μ0\mu\ge0, then μ\mu can be extended to a Radon measure (m,M)(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 XX is a locally compact Hausdorff topological space. A Radon measure on XX is a positive linear functional

μ:C c(X,R)R,\mu\colon C_c(X,\mathbf{R})\to\mathbf{R},

where C cC_c refers to the vector space of continuous compactly supported functions.

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

μ *(A)=inf{μ(χ B)BA,BisBorel}\mu^*(A)=\inf\{\mu(\chi_B)\mid B\supset A,\; B\; is\; Borel\}

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

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

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

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

Pushforwards of Radon measures

Definition

Suppose XX and YY are Hausdorff topological spaces and μ=(m,M)\mu=(m,M) is a Radon measure on XX. A map of sets H:XYH\colon X\to Y is Lusin μ\mu-measurable if for any compact KXK\subset X and ϵ>0\epsilon\gt0 there is a compact KKK'\subset K such that μ(KK)<ϵ\mu(K\setminus K')\lt\epsilon and the restriction of HH to KK' is continuous. We say that HH is μ\mu-proper if, in addition, every point of YY has a neighborhood whose inverse image is μ\mu-integrable.

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

Lusin μ\mu-measurable maps form a sheaf with respect to XX.

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

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

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

Definition

Suppose H:XYH\colon X\to Y is a μ\mu-proper map. The pushforward measure H *μH_*\mu is a Radon measure on YY defined as follows. Given a Borel subset BYB\subset Y, we set

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

This yields a Radon measure mm on YY.

Theorem

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

Properties

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

Examples

Most measures of interest in geometry are Radon. For example

See also

References

The canonical references on Radon measures are

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

Last revised on May 17, 2020 at 01:39:59. See the history of this page for a list of all contributions to it.