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Measure-theoretic properties of arbitrary point sets.

M Shiffman1

  • 1CALIFORNIA STATE COLLEGE, HAYWARD, CALIF. 94542.

Proceedings of the National Academy of Sciences of the United States of America
|September 1, 1970
PubMed
Summary

This study establishes complete conditions on interior and exterior Lebesgue measures for disjoint sets and their unions. It reveals that the average measure is subadditive, offering new insights into set measure properties.

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Area of Science:

  • Measure Theory
  • Real Analysis
  • Set Theory

Background:

  • Disjoint sets S(1) and S(2) have interior Lebesgue measure m(i)(S) and exterior Lebesgue measure m(e)(S).
  • Known inequalities include superadditivity of interior measure and subadditivity of exterior measure for disjoint sets.

Purpose of the Study:

  • To determine the complete set of conditions on the six quantities: m(i)(S1), m(e)(S1), m(i)(S2), m(e)(S2), m(i)(S1 U S2), and m(e)(S1 U S2).
  • To explore the properties of the average measure and linear combinations of interior and exterior measures.

Main Methods:

  • Derivation of a complete set of inequalities governing the six measure quantities.
  • Expression of measure variability using six independent nonnegative quantities.
  • Analysis of the subadditivity of the average measure, defined as (m(i)(S) + m(e)(S))/2.

Main Results:

  • A complete set of inequalities for the six measure quantities of disjoint sets and their unions has been obtained.
  • The variability of these measures can be fully described by six independent nonnegative quantities.
  • The average measure m(a)(S) is shown to be subadditive.

Conclusions:

  • The study provides a comprehensive characterization of the relationships between interior and exterior Lebesgue measures for set unions.
  • The findings offer a deeper understanding of measure properties beyond basic superadditivity and subadditivity.
  • The subadditivity of the average measure is an interesting emergent property with potential implications in further measure theory research.

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