The meaning of Asia in the Bible
(From Easton's Bible Dictionary)

is used to denote Proconsular Asia, a Roman province which embraced the western parts of Asia Minor, and of which Ephesus was the capital, in Acts 2:9; 6:9; 16:6; 19:10,22; 20:4,16,18, etc., and probably Asia Minor in Acts 19:26,27; 21:27; 24:18; 27:2. Proconsular Asia contained the seven churches of the Apocalypse (Revelation 1:11). The "chiefs of Asia" (Acts 19:31) were certain wealthy citizens who were annually elected to preside over the games and religious festivals of the several cities to which they belonged. Some of these "Asiarchs" were Paul's friends.

See more on the meaning of Asia in the Bible:
Asia {International Standard Bible Encyclopedia}
Asia {Hitchcock's Bible Name}

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