I'm following the attached lecture notes for proving "Convergence almost everywhere implies convergence in measure" with Egoroff's theorem.

Convergence almost everywhere implies convergence in mea- sure. Theorem 0.2. If fn - f a.e. and u(X ) 0, there exists N such that If (x) - fn(x)| N and all a: E E. So, for n > N, |f(:1:) fn(:1:)| can be greater than 6 only on (A \\ E) U (X \\ A). This means that M({56 \\ \\Mm) f(93)| > 6}) S MAW-*7) +M(X\\A) N. E] Egoroff's theorem (pointwise convergence is nearly uniform. Theorem 0.1. Suppose M(X ) 0, there exists a measurable subset E C X with u(X \\ E) f uniformly on E