Graduate Essentially bounded functions and simple functions

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Essentially bounded functions can be shown to be the uniform limit of simple functions under sigma-finite and positive measures. The proof involves demonstrating that the limit is measurable by using unions and intersections of measurable sets. To construct the simple functions, the domain is partitioned into compact pieces, and the range is divided into small intervals. Each interval is assigned a value based on the inverse image of that interval, creating simple functions that approximate the essentially bounded function within a specified range. The limit is uniform almost everywhere, acknowledging that the function is essentially bounded rather than strictly bounded.
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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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I need help. I forgot to indicate that the function is measurable also.
 
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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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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