Inspired by the first parenthetical sentence of Joel's answer to this question, I have the following question: is there any useful notion of measurability in the Baire space $\omega^\omega$?
Some initial thoughts:
Any countably additive, translation-invariant measure has to give measure 0 to any set of the form $\lbrace f: \sigma\prec f\rbrace.$ The same holds if we ask for a finitely additive translation-invariant measure which gives finite measure to the whole space.
There are several natural surjections $\omega^\omega\rightarrow 2^\omega$, and the latter space has a nice measure theory; so we could, for any such surjection $s$, define a "measure" given by $m_s(X)=m(s[X])$. However, it's unclear to me that any of these would be particularly interesting.
Even short of a decent "measure" on $\omega^\omega$ (and indeed, I believe none exists), we might still be able to talk about subsets of $\omega^\omega$ being "measurable" or "non-measurable" in terms of what sorts of pathologies they admit. For example, call $X\subseteq\omega^\omega$ definably measurable if $L(\mathbb{R}, X)\models $ "Every set is measurable." This is, however, a terrible notion of measurability: there are measure zero (and hence measurable) sets $X\subseteq 2^\omega$ such that $L(\mathbb{R}, X)$ contains a non-measurable set, so this is too restrictive.
Thoughts?