Real Analysis Exchange

Note on the Outer Measures of Images of Sets

F. S. Cater

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Abstract

Let $f$ be a real function on ${\mathbb R}$, let $\{I_v\}$ be a family of intervals covering a set $E$ such that $m(E \cap I_v) \ge m\bigl (f(E \cap I_v)\bigr )$ for each $I_v$. We prove that $m\bigl (f(E)\bigr ) \le 2 \cdot m(E)$. No coefficient smaller than $2$ will suffice here in general.

Article information

Source
Real Anal. Exchange, Volume 26, Number 2 (2000), 827-830.

Dates
First available in Project Euclid: 27 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1214571370

Mathematical Reviews number (MathSciNet)
MR1844396

Zentralblatt MATH identifier
1009.28006

Subjects
Primary: 28A12: Contents, measures, outer measures, capacities

Keywords
coverings Lebesgue outer measure

Citation

Cater, F. S. Note on the Outer Measures of Images of Sets. Real Anal. Exchange 26 (2000), no. 2, 827--830. https://projecteuclid.org/euclid.rae/1214571370


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References

  • H. Cullen, Introduction to General Topology, D. C. Heath, Boston, 1968 (Theorem 18.15).
  • E. Hewitt and K, Stromberg, Real and Abstract Analysis, Springer- Verlag, New York, 1965 (Exercises (17.25), (17.26), (17.27)).