Essentially bounded functions and simple functions

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  • Thread starter Shaji D R
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How to prove that essentially bounded functions are uniform limit of simple functions. Here measure is sigma finite and positive.
 

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  • #2
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I need help. I forgot to indicate that the function is measurable also.
 
  • #3
WWGD
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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
mathwonk
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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 fucntion is only essentially bounded and not bounded.
 

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