Proving the Existence of Rational Differences in a Measurable Set

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SUMMARY

The discussion centers on proving the existence of two elements in a measurable set with positive measure whose difference is a non-zero rational number. The user proposes defining a measurable set M and examining subsets M_n within intervals [n, n+1]. The key approach involves evaluating the measure of the union of translated sets q + M_n for rational numbers q within [-1, 1], ultimately aiming to demonstrate that M_n has measure zero, thereby contradicting the initial assumption.

PREREQUISITES
  • Understanding of measurable sets and their properties in real analysis.
  • Familiarity with rational numbers and their properties, specifically \mathbb{Q}\setminus \{0\}.
  • Knowledge of measure theory, including concepts like measure zero.
  • Basic skills in set theory and unions of sets.
NEXT STEPS
  • Study the properties of measurable sets in real analysis.
  • Learn about the implications of measure zero in the context of set theory.
  • Explore the concept of Lebesgue measure and its applications.
  • Investigate the role of rational numbers in measure theory and their impact on measurable sets.
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Mathematicians, students of real analysis, and researchers interested in measure theory and the properties of rational numbers within measurable sets.

modestoraton
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If i have a measurable set with positive measure, how do I prove that there are 2 elements who's difference is in Q~{0} (aka a rational number that isn't 0.
 
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Hi modestoraton! :smile:

Let M is measurable such that there are no two elements who's differences are in \mathbb{Q}\setminus \{0\}. Let M_n=M\cap[n,n+1].

Then perhaps you could evaluate the sum

\lambda\left(\bigcup_{q\in\mathbb{Q}\cap[-1,1]}{q+M_n}\right)

and show that Mn has measure zero.
 
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Thank you so much.
 

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