Measurable language which is not $\omega$-regular
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 $\{w\}$.