Measurable language which is not $\omega$-regular Let $\Sigma$ be a finite alphabet and let $\Sigma^\omega$ be the set of all infinite words over $\Sigma$. Consider 
$$
  d(x,y):=2^{-\min(n \in \Bbb N_0:x_n\neq y_n)}
$$
to be the metric on $\Sigma^\omega$ which makes the latter being the Cantor space. Denote by $\mathscr B(\Sigma^\omega)$ the Borel $\sigma$-algebra of this space, and let's call its elements measurable languages. It follows from Proposition 6 here that any $\omega$-regular language is measurable. I am looking for an example of a measurable language which is not $\omega$-regular, so any hints are appreciated.
A simple class of examples can be found by considering singleton languages $\{w\}$. These are measurable (Let $C_n(w)$ be the set of words agreeing with $w$ up to the $n$-th letter, then $\{w\}$ is the intersection of all $C_n(w)$). However, unless $w$ is eventually periodic, the language is not $\omega$-regular.

For a concrete example, consider the word $w=ababbabbba\dots$. If $w$ is accepted by a nondeterministic Büchi automaton $A$ with $k$ states, then while parsing the $n$-th block of $b$s, with $n&gt;k$, $A$ needs to visit some state twice, so we can construct another word accepted by $A$ by pumping this block suitably. Therefore no automaton recognizes $\{w\}$.