Are Simple Functions Dense in Bounded Borel Functions on a Compact Space?

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SUMMARY

Simple functions are proven to be dense in the space of bounded Borel functions on a compact space K when equipped with the supremum norm. The key argument involves demonstrating that for any bounded Borel function f and any ε > 0, there exists a sequence of simple functions f_n such that the supremum norm ||f_n - f|| approaches 0. This establishes the density of simple functions in the space B of bounded Borel functions.

PREREQUISITES
  • Understanding of compact spaces in topology
  • Familiarity with Borel functions and their properties
  • Knowledge of supremum norm and its implications in functional analysis
  • Basic concepts of convergence in normed spaces
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  • Study the properties of compact spaces and their implications in analysis
  • Learn about the construction and properties of Borel functions
  • Explore the concept of supremum norm in functional analysis
  • Investigate the proof techniques for density theorems in functional spaces
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Mathematicians, particularly those focused on functional analysis, topology, and measure theory, will benefit from this discussion. It is also relevant for students studying advanced calculus or real analysis.

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Let K be a compact space and let B be the space of bounded borel functions on K equipped with the supremum norm. Show that simple functions (i.e. functions attaining only a finite number of values) are dense in B.

Thanks
 
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Fermat said:
Let K be a compact space and let B be the space of bounded borel functions on K equipped with the supremum norm. Show that simple functions (i.e. functions attaining only a finite number of values) are dense in B.

Thanks

Hi Fermat,

Please make an effort and show us what you've done.

Thanks.
 
I know I need to show $||f_{n}-f||->0$ where the $f_{n}$ are simple but I don't know where to start
 

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