Give $(\Omega, \mathcal{F}, P)$ and $(X, \mathcal{B}(X))$, $\mathcal{M} :=\{\mu | \mu \text { is measure on }(X, \mathcal{B}(X))\}$, then random measure $\mathbf{P}$ can be defined as transition kernel $\mathbf{P}: \Omega \times \mathcal{B}(X) \to \mathbb{R}_+$ or a random element takes values in $\mathcal{M}$.
I know the definition of measure requires countable additivity. So does the definition of $\mathbf{P}$ imply $\mathbf{P}$ is finite additive or countably-additive?I mean,given measurable disjoint subset of $X, A_1,…,A_n$ , random variable $\mathbf{P}(\cup_n A_n) = \sum_n \mathbf{P}(A_n)$ i.e. sum of random variable $\mathbf{P}(A_n)$? If so, can $n$ be $\infty$?
My question comes from original paper of Dirichlet process:A Bayesian Analysis of Some Nonparametric Problems,
where it uses finite additivity(I am not sure finite additivity in the paper is assumption or the property of random measure)
Take Dirichlet process as example, does this process satisfy finite additive or countably-additive?
Best Answer
The usual definiton of a random measure requires countable additivity. If only finite additivity is assumed I think you should call your random measure a 'fintely additive random measure'.