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arXiv · 1506.01439

Integration and measures on the space of countable labelled graphs

Apoorva Khare·Bala Rajaratnam
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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 $\mathscr{G}(V)$.

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Apoorva Khare, Bala Rajaratnam. 2015-06-04. Integration and measures on the space of countable labelled graphs. https://arxiv.org/abs/1506.01439

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