Measures of Graphs on the Reals
نویسندگان
چکیده
This paper studies measure properties of graphs with infinitely many vertices. Let [0, 1] denote the real unit interval, and y be the collection of bijections taking [0, 1] onto itself. Given a graph G = ([0, \\,E) and / € & , define the f-representation of G to be the set Ef = {{f(x),f(y)):x,y e [0,1] and (x,y) e E} . Let p be 2-dimensional Lebesgue measure. Define the measure spectrum of G to be the set M(G) = {m 6 [0,1]:3/ e & with Ef measurable and pEf = m) . Our main result characterizes those subsets of reals that are the measure spectra of graphs.
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