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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Measures of graphs on the reals
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by Seth M. Malitz PDF
Proc. Amer. Math. Soc. 108 (1990), 77-87 Request permission

Abstract:

This paper studies measure properties of graphs with infinitely many vertices. Let $[0,1]$ denote the real unit interval, and $F$ be the collection of bijections taking $[0,1]$ onto itself. Given a graph $G = \left \langle {[0,1],E} \right \rangle$ and $f \in F$, define the $f$-representation of $G$ to be the set ${E_f} = \{ \langle {f(x),f(y)} \rangle :x,y \in [0,1]$ and $\langle {x,y} \rangle \in E\}$. Let $\mu$ be $2$-dimensional Lebesgue measure. Define the measure spectrum of $G$ to be the set $M(G) = \{ m \in [0,1]:\exists f \in F$ with ${E_f}$ measurable and $\mu {E_f} = m\}$. Our main result characterizes those subsets of reals that are the measure spectra of graphs.
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 108 (1990), 77-87
  • MSC: Primary 05C99; Secondary 03E05, 28A99
  • DOI: https://doi.org/10.1090/S0002-9939-1990-0982406-9
  • MathSciNet review: 982406