A Measurable Function not Borel Measurable

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SUMMARY

The discussion centers on identifying a Lebesgue measurable function that is not Borel measurable. Participants suggest using the characteristic function of a Lebesgue measurable set that is not Borel measurable as a concrete example. This approach highlights the distinction between Lebesgue and Borel measurability in the context of real analysis. The characteristic function serves as a practical tool for illustrating this concept.

PREREQUISITES
  • Understanding of Lebesgue measurable sets
  • Familiarity with Borel sets and their properties
  • Knowledge of characteristic functions in measure theory
  • Basic concepts of real analysis
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  • Research specific examples of Lebesgue measurable sets that are not Borel measurable
  • Study the properties and implications of characteristic functions in measure theory
  • Explore the differences between Lebesgue and Borel measurability in depth
  • Investigate advanced topics in real analysis related to measurable functions
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Mathematicians, students of real analysis, and researchers interested in measure theory and the nuances of measurability concepts.

zling110
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Most books have the famous example for a Lebesgue measurable set that is not Borel measurable. However, I am trying to find a Lebesgue measurable function that is not Borel measurable. Can anyone think of an example?
 
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Use the characteristic function of a set that's Lebesgue but not Borel.
 
dvs said:
Use the characteristic function of a set that's Lebesgue but not Borel.

thanks
 

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