Due to the history and development of measure and integration theory and different mathematical schools, there is a huge variety and inconsistency of definitions for concepts like tightness of a measure, regularity of a measure and even the definition for a measure $\mu : \mathcal{R} \to R$ on its own (including domain of definition such as a ring $\mathcal{R}$ (most often $\sigma$-algebra, $\sigma$-ring, $\delta$-ring or algebra) and target space $R$ (such as $[0, \infty]$ or $\mathbb{R}$ or $(-\infty, \infty]$ or a Banach space for vector measures).

There is also a huge variety of Riesz representation theorems between certain function spaces (bounded measurable functions $M_b(X)$, bounded continuous functions $C_b(X)$ (with sup-norm or some form of strict topology) and other classes of continuous functions such as $C_0(X)$ and $C_c(X)$ for locally compact domains $X$). Some representation theorems can be found in books like [Dunford Schwartz, "Linear Operators I"], [Bogachev, "Measure Theory II"], [Fremlin, "Measure Theory"] and many others. These representation theorems use some form of tightness or regularity, but the definitions are all different and in general not equivalent (which extremely slows down the "fast look-up" process.) For instance, one should distinguish between properties like (inner) closed-regular, (inner) compact-regular, outer-regular and so on. I would like to know, if there is a list somewhere that fixes a definition for all the various (useful) tightness and regularity properties and upon this summarizes all (or most of) the known Riesz representation theorems.

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    $\begingroup$ The list can be found here: ncatlab.org/nlab/show/Riesz%20representation%20theorem $\endgroup$ – Dmitri Pavlov Jul 17 '19 at 15:04
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    $\begingroup$ What's usually called a Riesz representation is one by Radon measures. There are at least two other types of theorems about representing functionals on spaces of functions by measures: Daniell (representation by countably additive measures) and A.D. Alexandroff (by finitely additive measures). $\endgroup$ – user95282 Jul 17 '19 at 19:23
  • $\begingroup$ @user95282 Problem is also that people define "Radon measure" differently and in general not equivalently. [Schwartz, "Radon measures"] has a nice discussion of some of the various definitions for a Radon measure. $\endgroup$ – yada Jul 22 '19 at 13:22
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    $\begingroup$ @DmitriPavlov The Riesz representation theorem in that link is only a very special case for $X$ a locally compact Hausdorff space and identification with $C_0(X)'$ and $C_c(X)'^+$. $\endgroup$ – yada Jul 22 '19 at 13:25

Such a list will always be based on subjective criteria but here is one suggestion, from a functional analytic rather than a probabilistic point of view. In my view the ingredients for an extension of the standard Riesz theorem for $C(K)$-spaces are

1) a space with structure (topology, metric, uniformity, $\sigma$-algebra,...);

2) vector spaces $E$ (of functions on the set—-the integrands) and $F$ (of measures). These are provided with topological structures which are compatible with the vector space one and under which they are complete;

3) $E$ and $F$ are in symmetric duality, i.e., each is naturally identifiable with the other‘s dual.

One would also expect these assignments of the vector spaces to the set to have natural functorial properties.

One can find such result for topological spaces (the duality between bounded, continuous functions and tight measures), uniform spaces (bounded, uniformly continuous functions and uniform measures (Pachl)) and measure spaces, i.e., sets with a $\sigma$-algebra (bounded, measurable functions and $\sigma$-additive measures).

In order to do this one has to extend the classical functional analytic structures to the categories of Saks and CoSaks spaces. However, this can be done in a unified manner, using Grothendieck‘s construction of ind and proj categories by applying it to the standard relevant categories (metric spaces, compacta, Banach spaces and the dual category—-Waelbroeck spaces).

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  • $\begingroup$ In the light of the above comments, I would add that I restricted myself to the case of bounded measures with approprate regularity properties ( $\sigma$-additivity, tightness, ...). All of the spaces of integrands I mentioned are also Banach spaces and their corresponding duals are much larger, incorporating finitely additive measures. I could also have carried on to consider the case of unbounded measures but this would have used much more space. In my experience it is the bounded (probability), regular measures which are of most interest in analysis and probability theory. $\endgroup$ – user131781 Jul 18 '19 at 6:12
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    $\begingroup$ On second thoughts, the following survey of results on characterising integrals as linear functionals is more in the spirit of your question—-„The Riesz-Radon-Fréchet problem on characterization of integrals“ by Zhakharov, Mikhalev and Rochonov in Russian Mathematical Surveys, 2010. $\endgroup$ – user131781 Jul 20 '19 at 6:24

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