# Why can't the complete class theorem be easily generalized to all locally-compact spaces?

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?

• I don't even see why the proof of Theorem 8.4.3. is true. In what topological vector space does the risk set live? – Michael Greinecker Sep 14 '18 at 13:01
• AFAIK the risk set is a subset of the infinite product R^Z, where Z is the compact set in question. It's assumed to be convex; the addition operation is straightforward. But as for the topology it's given, I'm actually not clear; I don't see Roberts talking about it, and the best guesses I can make are that it's some natural choice of operator norm or that it's the natural product topology. If it's the natural product topology, its compactness follows from Tychonoff's theorem, which would in fact answer my question: perhaps there is no version of Tychonoff's theorem for locally-compact spaces. – Billy Smith Sep 20 '18 at 12:44
• Actually, I don't even know whether Tychonoff's theorem would apply, since we've never assumed the set we're multiplying by itself Z times is compact, have we? You can have probability densities p(x) on compact sets that get unboundedly high - Dirac spikes - so AFAIK you can't prove that the risk set is compact, this way, you assume that the underlying space of possibilities is finite. My guess is you'd need to first provide another argument to show that the space of probability densities is compact, perhaps Prokhorov's theorem... then you can use Tychonoff? – Billy Smith Sep 20 '18 at 12:56