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Proceedings of the American Mathematical Society

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

 
 

 

Measures of graphs on the reals


Author: Seth M. Malitz
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
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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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Article copyright: © Copyright 1990 American Mathematical Society