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- W563670295 abstract "In this paper we develop a rigorous foundation for the study of integration and measures on the space $mathscr{G}(V)$ of all graphs defined on a countable labelled vertex set $V$. We first study several interrelated $sigma$-algebras and a large family of probability measures on graph space. We then focus on a dyadic Hamming distance function $left| cdot right|_{psi,2}$, which was very useful in the study of differentiation on $mathscr{G}(V)$. The function $left| cdot right|_{psi,2}$ is shown to be a Haar measure-preserving bijection from the subset of infinite graphs to the circle (with the Haar/Lebesgue measure), thereby naturally identifying the two spaces. As a consequence, we establish a change of variables formula that enables the transfer of the Riemann-Lebesgue theory on $mathbb{R}$ to graph space $mathscr{G}(V)$. This also complements previous work in which a theory of Newton-Leibnitz differentiation was transferred from the real line to $mathscr{G}(V)$ for countable $V$. Finally, we identify the Pontryagin dual of $mathscr{G}(V)$, and characterize the positive definite functions on $mathscr{G}(V)$." @default.
- W563670295 created "2016-06-24" @default.
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- W563670295 date "2015-06-04" @default.
- W563670295 modified "2023-09-27" @default.
- W563670295 title "Integration and measures on the space of countable labelled graphs" @default.
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