So I was reading Christian P. Robert's The Bayesian Choice, going through the constellation of results related to complete class theorems, and I don't see why all of them are necessary. In particular, in Theorem 8.4.3, a result is proven that I think comes from Wald. By invoking the Riesz(-Markov-Kakutani) representation theorem, and a related lemma about convex subsets of topological vector spaces, it is shown that every admissible decision rule for compact sets of parameters is a Bayes decision rule.
My confusion is, since the Riesz-Markov-Kakutani representation theorem applies to all locally-compact Hausdorff spaces rather than only compact spaces, why can't I immediately generalize this complete class theorem to all Euclidean spaces right off the bat? I see no obvious part of the proof that would fail if I simply go through the exact same proof, instead assuming that the space of parameters is locally compact, and then conclude that every admissible decision must be a generalized Bayes rule.
Of course, I know that this result is not actually true - the chapter essentially tells me that there are rules that can only be constructed as limits of Bayesian procedures rather than as generalized Bayes rules. But I don't see what would be wrong with this argument.
In general, trying to understand the constellation of related complete-class-theorem results is quite difficult, because you mostly have to wade through papers from 50+ years ago. I'd really appreciate some kind of reference that just gives the most general forms of all the results known. While on the topic, are there any (weak) variants of the complete class theorem with the audacious goal of categorizing admissible procedures for parameters that live in Polish spaces? Or is that just out of our mathematical reach as a species, or impossible?