Introducing the notion of a random measure, textbooks usually start with a locally compact second countable Hausdorff space. Where does this requirement come from?
I would like to have a motivation for this requirement. That is, I would like to define a random measure on a general measurable space $(X,\mathcal{B})$ simply as a kernel from a probability space to $(X,\mathcal{B}).$ Additional structure of $(X,\mathcal{B})$ should then be motivated by e.g. counterintuitive examples. For example, in the book of Last and Penrose (2017)
http://www.math.kit.edu/stoch/~last/seite/lectures_on_the_poisson_process/media/lastpenrose2017.pdf
Exercise 2.5 yields that point measures usually do not have the representation with a Dirac measures.
Are there other examples, (intuitive) motivations and reasons to use a locally compact (!) second countable (!) Hausdorff space?
