[2] Abe was born Abram-Aba Katsman in 1868 in Chișinău (then known as Kishinev), Bessarabia Governorate, Russian Empire.
[2][3][4][5] His father was Chaim Katzman and his mother was named Sura Bayla (née Goldman).
[4][6] He was from a musical family; his brother Philip Katzman played in opera orchestras in Moscow and Chișinău.
[8][9] By 1910, he was also apparently bandleader of a klezmer orchestra in Brooklyn, making him a contemporary of New York klezmer musicians from Romania and Bessarabia such as Max Leibowitz, Abe Schwartz, and Joseph Moskowitz, as well as Milu Lemisch in Philadelphia.
[5][16] Abe married his wife Rebecca (Rivke, née Sugarman/Zuckermann), daughter of a businessman from Akkerman, Bessarabia (now Bilhorod-Dnistrovskyi, Ukraine) in 1894 in Chișinău.