Proving Lebesgue Measure on R using Gdelta and Open Sets

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SUMMARY

The discussion focuses on proving the Lebesgue measure on the real numbers (R) using Gδ sets and open sets. It establishes that every measurable set in R can be expressed as the difference between a Gδ set and a set of measure zero. The participant initially attempted to simplify the problem by concentrating on Gδ sets and open sets, ultimately realizing that the solution did not require further decomposition into Gδ sets.

PREREQUISITES
  • Understanding of Lebesgue measure theory
  • Familiarity with Gδ sets and their properties
  • Knowledge of open sets in topology
  • Basic concepts of measure zero sets
NEXT STEPS
  • Study the properties of Gδ sets in detail
  • Explore the relationship between open sets and countable unions of intervals
  • Learn about the implications of measure zero sets in Lebesgue measure
  • Investigate advanced topics in measure theory, such as σ-algebras and measurable functions
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Mathematicians, students of real analysis, and anyone interested in measure theory and its applications in topology.

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Homework Statement


[PLAIN]http://www.album.com.hk/d/1546698-2/Untitled-1.jpg


Homework Equations


Every measurable set in R can be written as a difference of Gdelta set and a set of measure zero.
Every open set in R is just a countable union of disjoint intervals.

The Attempt at a Solution


Basically, I have reduced the problem into only concerning Gdelta sets (because sets of measure zero do not really matter that much), then I have further decomposed it into just a problem concerning only of open sets(if assuming it is true for open sets, I think I can prove it for Gdelta sets). So now I am hopelessly stuck, I have been thinking that every open set is just a countable union of disjoint interval, but I have no clue how to prove it for open sets.
 
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nevermind, I solved it, the solution isn't as complicated as I thought, I do not need to decompose it into Gdelta sets.
 

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