If a batted ball hits fair and bounces over the foul line between home plate and 1st base and the 1st baseman catches the ball in foul territory before it hits the ground is it fair or foul?

If a batted ball lands in fair territory but then crosses the foul line BEFORE passing or touching first or third base, then it is a FOUL ball.
If a batted ball lands in fair territory between first and third base, bounces in the air, and crosses into foul territory before passing first or third base, it is a foul ball, even if caught on the bounce in foul territory by the first or third baseman.