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.



List of all known Riesz representation theorems
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).


