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

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Discussion Overview

The discussion revolves around the question of whether simple functions are dense in the space of bounded Borel functions on a compact space, specifically under the supremum norm. Participants are exploring the theoretical aspects of this topic, including the necessary conditions and steps to demonstrate this property.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant requests a demonstration of the density of simple functions in the space of bounded Borel functions, indicating a need for a proof.
  • Another participant expresses uncertainty about how to begin the proof, specifically mentioning the requirement to show that the norm difference between a sequence of simple functions and a bounded Borel function approaches zero.
  • A later reply suggests that showing the density of simple functions would suffice to address the original question.

Areas of Agreement / Disagreement

There appears to be no consensus yet, as participants are still in the early stages of discussing the problem and have not reached any conclusions.

Contextual Notes

Participants have not yet outlined specific assumptions or definitions that may be relevant to the proof, and there are unresolved steps in the mathematical reasoning required to establish the density of simple functions.

Fermat1
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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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