Essentially bounded functions and simple functions

  • #1
How to prove that essentially bounded functions are uniform limit of simple functions. Here measure is sigma finite and positive.
  • #2
I need help. I forgot to indicate that the function is measurable also.
  • #3
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.
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  • #4
assume your function is bounded and divide up the range into small intervals. for each interval [a,b] take the functionm tohave value a on the inverse image of that interval...this gives you a simple function whichn lies within |b-a| of your function on that set...of course the limit is only uniform a.e. since the function is only essentially bounded and not bounded.

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