# Topological complexity of ultrafilters in $2^\kappa$ for uncountable $\kappa$

It is a well known fact that if $$\mathcal{F}$$ is a non-principal ultrafilter on $$\omega$$, then the set $$\{ \alpha \in 2^\omega : \alpha \in \mathcal{F}\}$$ (conflating binary strings with subsets of $$\omega$$) is not a Borel subset of $$2^\omega$$ with its standard product topology.

The proof of this that I am familiar with goes through showing that $$\mathcal{F} \subseteq 2^\omega$$ is not a measurable subset of $$2^\omega$$ by noting that if it were it would have density $$\frac{1}{2}$$ everywhere, contradicting the Lebesgue density theorem.

I am curious about the analogous statement with regards to ultrafilters on $$\kappa$$, considered as subsets of $$2^\kappa$$ with its compact product topology. I have difficulty imagining that a non-principal ultrafilter on $$2^\kappa$$ could be Borel (where by Borel I mean specifically an element of the $$\sigma$$-algebra generated by open sets, not just the $$\sigma$$-algebra generated by clopen sets), but I can't find a proof of this and the Lebesgue density theorem argument seems difficult to generalize to $$2^\kappa$$, even though there is a natural regular Borel measure on $$2^\kappa$$.

• Note: when considering $\kappa>\omega$, one can also consider the $\kappa^+$-complete algebra generated by open subsets, which contains more than just Borel subsets (say "$\kappa$-Borel") and ask if ultrafilters are $\kappa$-Borel. Or even ask about $2^\kappa$-complete. – YCor Oct 23 '20 at 6:40

A Borel ultrafilter would have the property of Baire. Therefore either $$\mathcal F$$ or its complement $$2^\kappa\setminus\mathcal F$$ is comeager relative to some basic open set. Since $$\mathcal F$$ is invariant under finite changes, this would mean $$\mathcal F$$ or its complement is comeager. Since $$\mathcal F$$ is the image of $$2^\kappa\setminus\mathcal F$$ complement under a homeomorphism of $$2^\kappa,$$ this means both sets are comeager and must therefore intersect, which is impossible.
Fun exercise: show that a Borel map $$2^\kappa\to[0,1]$$ that vanishes on finite subsets of $$\kappa$$ cannot be a finitely additive probability measure.