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.