Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\mathscr{G}(V)$ 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
$\mathscr{G}(V)$.

14
The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs