Definitions
1.NounA major league.
2.AdjectiveMajor league (pertaining to the highest level of professional sports)."He made a real big league play."
3.AdjectiveMajor league (significant, important).
4.AdverbTo a significant degree; on a large scale.
5.Adverb↑ Liam Stack (24 October 2016), “Yes, Trump Really Is Saying ‘Big League,’ not ‘Bigly,’ Linguists Say”, in The New York Times