# Essentially bounded functions and simple functions

• A
How to prove that essentially bounded functions are uniform limit of simple functions. Here measure is sigma finite and positive.

I need help. I forgot to indicate that the function is measurable also.

WWGD
Gold Member
Trick is usually to describe limit in terms of unions, intersections of measurable sets. I mean this in order to show that the limit is measurable. for the rest, partition your domain in "enough" (compact) pieces for the vertical intervals [n, n+1). I think I remember Wikipedia had a proof.

Last edited:
mathwonk