I am looking into the Wikipedia article with the topic Randomised decision rule.
In the "Definition and interpretation" section, I see the formula of randomized loss:
$$L(\theta,d^*)=\int_{A\in\mathcal{A}}d^*(x,A)L(\theta,A)dA$$
where $\mathcal{A}$ is the action space.
What bothers me is that the action space can be anything, not necessarily a subset of real numbers, so it does not make senseto integrate with respect to an action. Can anybody help me to make sense out of this?
EDIT: After looking into several books, including Lehmann, I'm ashamed to admit that I still not understand the domain, codomain and the (joint?) measurability details of the above.