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).Comment: 15 pages, LaTe