Relations between various definitions of a Radon measure
The following various definitions of a Radon measure seem to be given for the Borel sigma algebra of different types of topological spaces: general, Hausdorff, locally compact, or locally compact Hausdorff.
I was wondering if the definitions are related in some way?
Can these definitions or most of them be unified?
References are appreciated! Thanks and regards!
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From Measure Theory, Volumes 1-2 by Vladimir I. Bogachev
Let $X$ be a topological space. A Borel measure $\mu$ on $X$ is called a Radon measure if for every $B$ in $B(X)$ and $ε>0$, there exists a compact set $K_ε ⊂ B$ such that $|\mu|(B - K_ε) <ε$.
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From Wikipedia:
On the Borel $σ$-algebra of a Hausdorff topological space $X$, a measure is called a Radon measure if it is
- locally finite, and
- inner regular.
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From ncatlab
If $X$ is a locally compact Hausdorff topological space, a Radon measure on $X$ is a Borel measure on $X$ that is
- finite on all compact sets,
- outer regular on all Borel sets, and
- inner regular on open sets.
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From planetmath
Let $X$ be a Hausdorff space. A Borel measure $\mu$ on $X$ is said to be a Radon measure if it is:
- finite on compact sets,
- inner regular (tight).
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From Wikipedia's Radon measures on locally compact spaces
When the underlying measure space is a locally compact topological space, the definition of a Radon measure can be expressed in terms of continuous linear functionals on the space of continuous functions with compact support.
Solution 1:
I will concentrate on comparing (3) and (4). The definition (1) is meant for finite signed measures, whereas all the other definitions are meant for arbitrary positive measures; (1) is equivalent to (4) in the case of finite positive measures. (2) appears to be equivalent to (4) ["locally finite" can mean "finite on compact sets", although it is sometimes taken to mean "finite on the elements of some topological basis"; these are equivalent in the LCH (locally compact Hausdorff) case]. Finally, (5) does not appear to be a definition at all, but rather a description of a definition.
Now then,
i) In the case of a second countable LCH space, every locally finite measure satisfies both (3) and (4) (Theorem 7.8 of [1]). This is the most commonly considered scenario in applications, which is why almost no one bothers to carefully sort out the differences between the different definitions.
ii) In the case of a sigma-compact LCH space, (3) and (4) are equivalent. The forward direction is Corollary 7.6 of [1]; the backwards direction follows from the forward direction together with (iv) below (but I'm sure there is an easier proof).
iii) (3) and (4) are not equivalent in general, even for LCH metrizable spaces (Exercise 7.12 of [1]).
iv) In an LCH space, there is a bijection between
A) measures satisfying (3),
B) measures satisfying (4), and
C) positive linear functionals on the space of continuous functions with compact support.
(The Riesz representation theorem gives either (A)<->(C) or (B)<->(C), depending on where you look; (A)<->(B) is in the Schwarz book mentioned by Joe Lucke; see also Exercise 7.14 of [1])
[1] G. B. Folland, Real Analysis: Modern Techniques and Their Applications
Note: In [1], "Radon" refers to measures satisfying (3).
Solution 2:
Schwartz (Radon measures on arbitrary topological spaces and cylindrical measures, 1973) defines Radon measures as comprising two measures. The first is the measure given in version 3 above and the second is the essential measure defined as locally finite, tight measure He then shows that each can generate the other. On LCH spaces, version 3 equivalent to version 5. Prinz (Regularity of Riesz measures, 1986, Proc Amer Math Soc) calls version 3 a "Riesz" measure and the locally finite, tight version a "Radon" measure and refers to Schwartz to give their duality.