How can I prove the following result?
Uniform convergence almost everywhere implies convergence almost uniformly.
I know Egorov's theorem which says pointwise convergence almost everywhere implies convergence almost uniformly when measure of domain is finite.
Also, uniform convergence almost everywhere implies pointwise convergence almost everywhere. But Egorov's theorem applies only for finite measure domain.
That means from the result written above, we can conclude that finite measure domain is not required in case of uniform convergence almost everywhere.