  Choquet theorems (1954) on integral representation for capacities are
fundamental to probability theory. They inspired a growing body of research
into different approaches and generalizations of Choquet's results by many
other researchers. Notably Math\'{e}ron's work (1975) on distributions over the
space of closed subsets has led to further advancements in the theory of random
sets. This paper was inspired by the work of Norberg (1989) who generalized
Choquet's results to distributions over domains. While Choquet's original
theorems were obtained for locally compact Hausdorff (LCH) spaces, both
Math\'{e}ron's and Norberg's depend on the assumption of separability in their
application of the Carath\'{e}odory's method. Our Radon measure approach
differs from the work of Math\'{e}ron and Norberg, in that it does not require
separability. This investigation naturally leads to the introduction of finite
and locally finite valuations, which allows us to characterize finite and
locally finite random sets in terms of capacities on the class of compact
subsets. Finally, the treatment of L\'{e}vy exponent by Math\'{e}ron and
Norberg is revisited, and the notion of exponential valuation is proposed for
the representation of general Poisson processes.
