TITLE: What is a random measure?
QUESTION [9 upvotes]: What is a random measure?
The Wikipedia article is quite confusing and used to formulate only a random counting measure instead of just the random measure. I'm working on the Dirichlet process and with the more in-depth articles they start with the notion of a random measure, but there is not much intuition given. Are there perhaps multiple definitions?
From what I understand it is a mapping from a probability space $(\Omega,\mathbb{F},\mathbb{P})$ to a measurable space $(S,\Sigma)$ just as a $(S,\Sigma)$-valued random variable. 
The difference is that it is a specific $(S,\Sigma)$-valued random variable, namely one that is set of normal ($\mathbb{R}$-valued) random variables. 
Is this correct? Are there some good examples of different types of random measures?

REPLY [8 votes]: As you say, a random measure is a measure-valued RV.  You need a sigma algebra structure on the space of measures for this.  Often times, the measures will be defined on a separable metric space, and you will be able to use the Prokhorov topology.
An "equivalent" (i.e. "naturally identified") understanding of random measures in the case mentioned above is to say the following: $\alpha: \Omega\times\Sigma\rightarrow [0, 1]$ is a random measure if fixing the left coordinate always gives you a probability measure, and fixing the right coordinate always a measurable map. (See, for instance, Kechris' Classical Descriptive Set Theory Ch. 17)
Random measures are used when talking about regular conditional distributions and exchangeability.