  The main goal of this paper is to build a measurable analogue to the theory
of weighted networks on infinite graphs. Our basic setting is an infinite
$\sigma$-finite measure space $(V, \mathcal B, \mu)$ and a symmetric measure
$\rho$ on $(V\times V, \mathcal B\times \mathcal B)$ supported by a measurable
symmetric subset $E\subset V\times V$. This applies to such diverse areas as
optimization, graphons (limits of finite graphs), symbolic dynamics, measurable
equivalence relations, to determinantal processes, to jump-processes; and it
extends earlier studies of infinite graphs $G = (V, E)$ which are endowed with
a symmetric weight function $c_{xy}$ defined on the set of edges $E$. As in the
theory of weighted networks, we consider the Hilbert spaces $L^2(\mu),
L^2(c\mu)$ and define two other Hilbert spaces, the dissipation space $Diss$
and finite energy space $\mathcal H_E$. Our main results include a number of
explicit spectral theoretic and potential theoretic theorems that apply to two
realizations of Laplace operators, and the associated jump-diffusion
semigroups, one in $L^2(\mu)$, and, the second, its counterpart in $\mathcal
H_E$. We show in particular that it is the second setting (the energy-Hilbert
space and the dissipation Hilbert space) which is needed in a detailed study of
transient Markov processes.
