Essentially bounded functions and simple functions

In summary, to prove that essentially bounded functions are uniform limit of simple functions, the function must be measurable and the measure must be sigma finite and positive. A trick to do this is to describe the limit in terms of unions and intersections of measurable sets. Wikipedia also has a proof for this. Additionally, if the function is bounded, it can be divided into small intervals and a simple function can be created by assigning a value to the inverse image of each interval. This simple function will lie within a small distance of the original function on that set, but the limit is only uniform almost everywhere since the function is only essentially bounded and not bounded.
  • #1
Shaji D R
19
0
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
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.
 

1. What is an essentially bounded function?

An essentially bounded function is a function that is bounded almost everywhere, meaning that it is bounded everywhere except on a set of measure zero. This means that the function has a finite upper and lower bound on all but a small portion of its domain.

2. What is the difference between an essentially bounded function and a bounded function?

A bounded function has a finite upper and lower bound on its entire domain, while an essentially bounded function has a finite upper and lower bound on all but a small portion of its domain. Essentially bounded functions can still have unbounded behavior on a set of measure zero.

3. What is a simple function?

A simple function is a function that takes on a finite number of values over its domain. It can be thought of as a piecewise constant function, where each piece is a constant value over a specific interval. Simple functions are commonly used in measure theory to approximate more complex functions.

4. How are simple functions used in measure theory?

Simple functions are used in measure theory as approximations for more complex functions. They can be used to prove important theorems, such as the Monotone Convergence Theorem and the Dominated Convergence Theorem. They also play a crucial role in defining the integral of a function.

5. What is the significance of essentially bounded functions and simple functions in mathematics?

Essentially bounded functions and simple functions are important concepts in analysis and measure theory. They allow us to study and approximate more complex functions in a rigorous and systematic way. They also play a crucial role in many important theorems and concepts, such as the Lebesgue integral and convergence theorems.

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