Measurability of an $\omega$-regular language
An $\omega$-regular language is actually quite low in the Borel hierarchy (inside $\Delta_3$), a result due to 

R. McNaughton, Testing and generating infinite sequences by a finite automaton, Information and Control 9 (1966), 521-530.  

For a proof and more details, you can look at Chapter 3 of the following book

D. Perrin et J.-É. Pin, Infinite words, Pure and Applied Mathematics Vol 141, Elsevier, 2004, ISBN 0-12-532111-2.