Integration and Measures on the Space of Countable Labelled Graphs

In this paper we develop a rigorous foundation for the study of integration and measures on the space G (V ) of all graphs defined on a countable labelled vertex set V . We first study several interrelated σ-algebras and a large family of probability measures on graph space. We then focus on a “dyadic” Hamming distance function ‖·‖ψ,2, which was very useful in the study of differentiation on G (V ). The function ‖·‖ψ,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 R to graph space G (V ). This also complements previous work in which a theory of Newton-Leibnitz differentiation was transferred from the real line to G (V ) for countable V . Finally, we identify the Pontryagin dual of G (V ), and characterize the positive definite functions on G (V ).