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Proceedings of the American Mathematical Society

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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
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

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