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# Sheaves of measures and KMS-weights on topological graph algebras

Jonas Eidesen  Department of Mathematics, University of Oslo, Norway
<EMAIL_ADDRESS>
(Date: October 5th 2023)

###### Abstract.

We show that the collection of regular Borel measures on a second-countable
locally compact Hausdorff space has the structure of a sheaf. With this we
give an alternate description of the pullback of a regular Borel measure along
a local homeomorphism. We are able to use these tools to give a description of
the KMSβ-weights for the gauge-action on the graph $C^{*}$-algebra of a
second-countable topological graph in terms of sub-invariant measures on the
vertex space of said topological graph.

##  1\. Introduction

Topological graphs give a setting for studying a large family of
$C^{*}$-algebras, among others it generalizes the existing theory of graph
algebras and gives a new setting to study homeomorphism $C^{*}$-algebras. A
lot of the theory was first developed by Katsura in [Katsura2003]. Katsura
associated a $C^{*}$-algebra to the quadruple $E=(E^{0},E^{1},r,s)$ where
$E^{0}$ and $E^{1}$ are locally compact Hausdorff spaces, $s:E^{1}\rightarrow
E^{0}$ is a local homeomorphism and $r:E^{1}\rightarrow E^{0}$ is continuous.
It was shown by Yeend in [Yeend2006, Yeend2007] that the $C^{*}$-algebras
constructed from topological graphs admit a groupoid model. This is done by
constructing the boundary path space $\partial E$ and constructing a monoidal
action on this space from the natural numbers $\mathbb{N}$ via the _backwards
shift map_ $\sigma:\partial E\setminus E^{0}\rightarrow\partial E$. With this
one can consider the Deaconu-Renault groupoid of this action.

Groupoid $C^{*}$-algebras have been a fruitful class of $C^{*}$-algebras when
studying dynamical systems and KMSβ-states. Given a groupoid $\mathcal{G}$ and
a continuous groupoid homomorphism $c:\mathcal{G}\rightarrow\mathbb{R}$, one
is able to construct a $C^{*}$-dynamical system
$(C^{*}(\mathcal{G}),\mathbb{R},\alpha^{c})$ using Pontryagin duality. A lot
of work was initiated by Renault in [Renault1980] where he is able to give a
description of the KMSβ-states for $\alpha^{c}$ in the case where
$\mathcal{G}$ is a locally compact Hausdorff étale groupoid. He shows that
each KMSβ-state for $\alpha^{c}$ restricts to a quasi-invariant probability
measure on the unit space $\mathcal{G}^{(0)}$ with Radon-Nikodym derivative
$e^{-\beta c}$. When the groupoid is principal he is also able to show that
these KMSβ-states are in fact all the KMSβ-states for $\alpha^{c}$.

In [Neshveyev2013] Neshveyev is able to generalize Renault’s description of
KMSβ-states for $\alpha^{c}$ on principal étale groupoids to the non-principal
case. Christensen is able to further generalize Neshveyev’s results to give a
description of KMSβ-_weights_ for $\alpha^{c}$ in [Christensen2023]. It is
worth mentioning that there exist $C^{*}$-dynamical systems for which there
does not exist any KMSβ-states, but there does exist KMSβ-weights. Hence, it
is worth it to strive for describing not only the KMSβ-states of a
$C^{*}$-dynamical system, but also the KMSβ-weights. We give an example of
such a situation at the end of this paper, c.f. Example 5.9.

Schafhauser initiated the study of tracial states on $C^{*}$-algebras
associated to topological graphs in [Schafhauser2018]. The $C^{*}$-algebra of
a topological graph comes equipped with a natural action from the circle
group, generally called the _gauge-action_. Schafhauser was in particular
interested in the tracial states which are gauge-invariant, and he was able to
give a description of these tracial states both in terms of invariant
probability measures on the boundary path space $\partial E$ and vertex-
invariant probability measures on the vertex space $E^{0}$. Schafhauser was
unable to completely characterize the gauge-invariant tracial states but
conjectured that for free topological graphs, every tracial state should be
gauge-invariant.

In [Christensen2022] Christensen uses the program developed by Schafhauser
along with his results from [Christensen2023] to study tracial _weights_ on
$C^{*}$-algebras associated to _second-countable_ topological graphs. He is in
particular able to give a description of the gauge-invariant tracial weights,
and proves Schafhauser’s conjecture to be true.

We realized that the techniques developed by Schafhauser in [Schafhauser2018]
and Christensen in [Christensen2022] can be used to study KMSβ-weights for the
gauge-action on $C^{*}$-algebras associated to second-countable topological
graphs.

In addition to studying the structure of KMSβ-weights for the gauge-action on
the $C^{*}$-algebra of a second-countable topological graph, we became
interested in the purely measure-theoretic questions that arose in our
studies. In particular, there are a lot of results in functional analysis that
are of the form ”this set of functionals is in bijection with this set of
measures”. Riesz representation Theorem is the archetypical example of this
type of result. In reality, we almost never use these measures explicitly,
more often then not we only concern ourselves with the integral over these
measures and work with these integrals. There should however be a way to work
with these measures directly instead of constantly appealing to the duality
between functionals and measures.

The motivating example that led us towards studying this deeper was the idea
of pulling back measures. The existing idea has the following setup: let
$\varphi:X\rightarrow Y$ be a local homeomorphism between locally compact
Hausdorff spaces and $\mu$ be a regular Borel measure on $Y$. Then there exist
a unique regular Borel measure $\varphi^{*}\mu$ on $X$ such that

(1.1) $\int_{X}fd\varphi^{*}\mu=\int_{Y}\sum_{x\in\varphi^{-1}(y)}f(x)d\mu(y)$

for all $f\in C_{c}(X)$.

It isn’t immediately clear from this formula what the measure $\varphi^{*}\mu$
actually _measures_. Our goal became to obtain a definition of the pullback
which preserves the geometric intuition one would hope to have. In particular,
since $\varphi$ is a _local_ homeomorphism one would expect that
$\varphi^{*}\mu$ would _locally_ have the same behavior as $\mu$. This is made
precise in Proposition 3.7.

We use this new description of the pullback (of a regular Borel measure) to
give a complete description of the KMSβ-weights for the gauge action of the
graph $C^{*}$-algebra of a second-countable topological graph.

###  1.1. Outline

In Section 2 we go through all the necessary prerequisites about sheaf-theory,
groupoids and groupoid $C^{*}$-algebras as well as topological graphs. We
particularly sketch the construction of the graph groupoid and the gauge-
action on the graph $C^{*}$-algebra. The fact that the $C^{*}$-algebra of a
topological graph admits a groupoid model (due to Yeend, c.f. [Yeend2006,
Yeend2007]) allows us to use Renault’s, Neshveyev’s and Christensen’s
classification of KMSβ-states and -weights on groupoids, (c.f. [Renault1980,
Neshveyev2013, Christensen2023]) to get a similar classification of KMSβ-
states and -weights for the gauge-action on the $C^{*}$-algebra of a second-
countable topological graph.

In Section 3 we introduce the functors $\mathcal{M}$ and
$\mathcal{M}_{\text{reg}}$ from the topology on a space $X$ (viewed as a
poset-category) to the category of sets, sending an open subset to the
collection of all Borel measures and regular Borel measures on that open
subset respectively. We show that for arbitrary topological spaces, these
functors have the structure of presheaves ($\mathcal{M}_{\text{reg}}$ has the
structure of a sub-presheaf of $\mathcal{M}$). For second-countable spaces we
show that $\mathcal{M}$ has the structure of a sheaf, and for second-countable
locally compact Hausdorff spaces, $\mathcal{M}_{\text{reg}}$ has the structure
of a sheaf. We use this to give a definition of the pullback of a regular
Borel measure along a local homeomorphism, which coincides with the pullback
as defined through Equation 1.1, c.f. Proposition 3.7 and Proposition 3.10.

In Section 4 we translate the results of Neshveyev and Christensen, (c.f.
[Neshveyev2013, Christensen2023]) from the language of groupoid
$C^{*}$-algebras to the language of topological graph $C^{*}$-algebras. In
particular, we introduce _$\beta$ -quasi-invariant measures_, which are
measures on the boundary path space of a second-countable topological graph
that turn out to be the measures that correspond to the KMSβ-weights for the
gauge action on the $C^{*}$-algebra of a second-countable topological graph,
c.f. Proposition 4.3.

In Section 5 we introduce _$\beta$ -sub-invariant measures_, which are
measures on the vertex space of a second-countable topological graph. We prove
that there is an affine bijection between the $\beta$-quasi-invariant measures
of a second-countable topological graph and the $\beta$-sub-invariant
measures. Since the vertex space of a second-countable topological graph will
in general be easier to get a grasp of then the boundary path space of a
second-countable topological graph, this bijection allows one to, for example,
compute the KMS-spectrum of the graph $C^{*}$-algebra (in regard to the gauge-
action).

To keep things more organized in this paper we mark the end of definitions
with the symbol $\heartsuit$, and we end remarks and examples with the symbol
$\diamondsuit$.

###  1.2. Acknowledgments

The contents of this paper is based on the authors master’s thesis, written at
the University of Oslo, supervised by Nadia Larsen.

##  2\. Prerequisites

###  2.1. Sheaves

Since sheaf theory is not the most standard tool in functional analysis it
will be useful to recall the definitions here. See [Hartshorne1977] for a
thorough introduction in the setting of algebraic geometry.

Let $X$ be a topological space and denote its topology by $\mathcal{T}$. A
_presheaf of sets on $X$_ is a contravariant functor
$\mathcal{F}:\mathcal{T}\rightarrow\mathbf{Sets}$ where $\mathcal{T}$ is to be
understood as a poset-category ordered by inclusion. For $U,V\in\mathcal{T}$
the elements $s\in\mathcal{F}(V)$ are called _sections_ and if $U\subset V$ we
call the function $\rho^{V}_{U}:=\mathcal{F}(U\subset V)$ for a _restriction_
, and we use the notation $\rho^{V}_{U}(s)=s|_{U}$.

A _sheaf of sets on $X$_ is a presheaf of sets on $X$, $\mathcal{F}$,
satisfying the two following axioms for any open subset $W\subset X$ and any
open cover $\mathcal{U}$ of $W$:

  * (_Locality_)

If $s,t\in\mathcal{F}(W)$ are such that $s|_{U}=t|_{U}$ for all
$U\in\mathcal{U}$ we have that $s=t$.

  * (_Gluing_)

If we have a family of sections $\\{s_{U}\\}_{U\in\mathcal{U}}$ such that
$s_{U}\in\mathcal{F}(U)$ and $s_{U}|_{U\cap V}=s_{V}|_{U\cap V}$ for every
pair $U,V\in\mathcal{U}$, then there exists a section $s\in\mathcal{F}(W)$
such that $s|_{U}=s_{U}$ for all $U\in\mathcal{U}$.

It is clear that the gluing axiom gives the existence of a global section
whenever there are local sections that agree on intersections, and the
locality axiom gives uniqueness of this global section. A priori one might
also believe that this global section is dependent on the open cover
$\mathcal{U}$, it is however easy to show that this is not the case.

If $\mathcal{H}$ is also a (pre)sheaf of sets on $X$ such that
$\mathcal{H}(U)\subset\mathcal{F}(U)$ and the diagram

$\textstyle{\mathcal{F}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}(U)}$$\textstyle{\mathcal{H}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{H}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

commutes for each pair of open subsets $U,V\subset X$ such that $U\subset V$,
we say that $\mathcal{H}$ is a _sub-(pre)sheaf_ of $\mathcal{F}$. Note that
the vertical arrows in this diagram are inclusions and the horizontal arrows
are the restrictions in $\mathcal{F}$ and $\mathcal{H}$ respectively.

It is worth recalling that a (pre)sheaf can take values in any category, not
only $\mathbf{Sets}$. A particularly useful example is when a (pre)sheaf takes
values in $\mathbf{Ab}$, the category of abelian groups. In this case, one has
access to the tools of homological algebra, thus it is possible to define
sheaf-cohomology, which is a powerful tool in many settings.

###  2.2. Étale groupoids and their $C^{*}$-algebras

For a thorough introduction to this topic see the work done by Renault in
[Renault1980]. We define a groupoid $\mathcal{G}$ to be a small category where
each morphism is invertible. We denote the range map (codomain map) by
$r:\mathcal{G}\rightarrow\mathcal{G}$ and the source map (domain map) by
$s:\mathcal{G}\rightarrow\mathcal{G}$. The set of objects of $\mathcal{G}$ is
denoted by $\mathcal{G}^{(0)}$ and will be referred to as the _unit space_. We
will generally use the letters $x,y,z\in\mathcal{G}$ to denote general
elements in the groupoid and the letters $u,v,w\in\mathcal{G}^{(0)}$ to denote
units. For a unit $u\in\mathcal{G}^{(0)}$ we define
$\mathcal{G}_{u}=s^{-1}(u)$ and $\mathcal{G}^{u}=r^{-1}(u)$, and denote by
$\mathcal{G}_{u}^{u}=\mathcal{G}_{u}\cap\mathcal{G}^{u}$ the _isotropy
subgroup at $u$_. We will only consider groupoids which are second-countable
locally compact Hausdorff and étale, where étale means that both the range map
and source map are local homeomorphisms. To avoid cluttered terminology these
groupoids will simply be referred to as _étale groupoids_. An open subset $W$
of an étale groupoid $\mathcal{G}$ is called a _bisection_ if both $r|_{W}$
and $s|_{W}$ are injective. An important fact is that every étale groupoid has
a countable base of bisections.

We construct a $*$-algebra from an étale groupoid $\mathcal{G}$ by looking at
the complex vector space $C_{c}(\mathcal{G})$ with the convolution product
$f*g$ of two functions $f,g\in C_{c}(\mathcal{G})$ defined by the equation

$(f*g)(x)=\sum_{y\in\mathcal{G}^{r(x)}}f(y)\,g(y^{-1}x)$

for any $x\in\mathcal{G}$, and involution defined by
$f^{*}(x)=\overline{f(x^{-1})}$. The full $C^{*}$-norm on $C_{c}(\mathcal{G})$
is defined by the equation

$\|f\|=\sup\\{\|\pi(f)\|\ |\ \pi\text{ is a }*\text{-representation of
}C_{c}(\mathcal{G})\\}$

for $f\in C_{c}(\mathcal{G})$, and the full groupoid $C^{*}$-algebra is
defined as the completion of $C_{c}(\mathcal{G})$ in this norm, namely

$C^{*}(\mathcal{G})=\overline{C_{c}(\mathcal{G})}^{\|\,\cdot\,\|}.$

###  2.3. Dynamical systems

For an introduction in $C^{*}$-dynamical systems see for example
[Bratteli1987]. A _$C^{*}$ -dynamical system_ is a triple
$(\mathcal{A},G,\alpha)$ with $\mathcal{A}$ being a $C^{*}$-algebra, $G$ a
locally compact group and $\alpha$ a strongly continuous map
$\alpha:G\rightarrow\operatorname{Aut}(\mathcal{A})$ satisfying
$\alpha_{e}=id_{\mathcal{A}}$ and $\alpha_{g}\alpha_{h}=\alpha_{gh}$, where
$e\in G$ is the identity element and $g,h\in G$. By strongly continuous we
mean that the map $g\mapsto\alpha_{g}(a)$ is norm-continuous for all
$a\in\mathcal{A}$.

We will in particular be interested in the case where $G=\mathbb{R}$. In this
case it will be possible to analytically continue the dynamics of $\alpha$
from $\mathbb{R}$ to $\mathbb{C}$. Hence, let
$(\mathcal{A},\mathbb{R},\alpha)$ be a $C^{*}$-dynamical system. For
$z\in\mathbb{C}$ with $\operatorname{Im}(z)\geq 0$ we define the set $S(z)$ to
be the horizontal strip $S(z)=\\{w\in\mathbb{C}\ |\
\operatorname{Im}(w)\in[0,\operatorname{Im}(z)]\\}$. When
$\operatorname{Im}(z)\leq 0$ we define $S(z)$ similarly:
$S(z)=\\{w\in\mathbb{C}\ |\
\operatorname{Im}(w)\in[\operatorname{Im}(z),0]\\}$.

We define the set $D(\alpha_{z})\subset\mathcal{A}$ to be the elements
$a\in\mathcal{A}$ such that there exists a continuous function
$f:S(z)\rightarrow\mathcal{A}$ which is _analytic_ on the interior of $S(z)$
such that $f(t)=\alpha_{t}(a)$ for all $t\in\mathbb{R}$. We define
$\alpha_{z}(a)=f(z)$ for $z\in S(z)$. Analytic in this context means that the
composition $\varphi\circ f$ is holomorphic on the interior of $S(z)$ for all
$\varphi\in\mathcal{A}^{*}$. We say that an element $a\in\mathcal{A}$ is
_analytic for $\alpha$_ if there exists an entire function
$f:\mathbb{C}\rightarrow\mathcal{A}$ such that $f(t)=\alpha_{t}(a)$ for all
$t\in\mathbb{R}$.

With the basic concepts about $C^{*}$-dynamical systems established we turn
our attention towards _weights_ , [Pedersen1979] is a nice resource for
learning the basics about weights. Let $\mathcal{A}$ be a $C^{*}$-algebra and
denote by $\mathcal{A}_{+}$ the convex cone of positive elements in
$\mathcal{A}$. A _weight_ on $\mathcal{A}$ is a map
$\psi:\mathcal{A}_{+}\rightarrow[0,\infty]$ such that for all
$a,b\in\mathcal{A}_{+}$ and $\lambda\geq 0$ we have that

  * •

$\psi(a+b)=\psi(a)+\psi(b)$ and

  * •

$\psi(\lambda a)=\lambda\psi(a)$.

Denote by $\mathcal{A}_{+}^{\psi}$ the set of all positive elements
$a\in\mathcal{A}_{+}$ for which $\psi(a)<\infty$. Since the positive elements
of a $C^{*}$-algebra span the entire $C^{*}$-algebra we define
$\mathcal{A}^{\psi}=\operatorname{span}(\mathcal{A}_{+}^{\psi})$, and we may
extend $\psi$ uniquely to a positive functional on $\mathcal{A}^{\psi}$ in the
obvious way. (This is done in detail in [Pedersen1979]*Lemma 5.1.2).

We call a weight $\psi$

  * •

_densely defined_ if $\mathcal{A}_{+}^{\psi}$ is dense in $\mathcal{A}_{+}$,

  * •

_lower semi-continuous_ if $\\{a\in\mathcal{A}_{+}\ |\ \psi(a)\leq\lambda\\}$
is closed for all $\lambda\geq 0$, and

  * •

_proper_ if it is densely defined and lower semi-continuous.

Given a $C^{*}$-dynamical system, it is of interest to ask which weights are
invariant under this dynamic. The theory of KMSβ-weights gives an answer to
this question and was first introduced by Combes in [Combes1971]. We will
however use the following description of KMSβ-weights, c.f.
[Kustermans1997]*Definition 2.8. Let $(\mathcal{A},\mathbb{R},\alpha)$ be a
$C^{*}$-dynamical system and $\beta\in\mathbb{R}$. We call a weight $\psi$ on
$\mathcal{A}$ a _KMS β-weight for $\alpha$_ if it is a proper weight
satisfying

  * •

$\psi\circ\alpha_{t}=\psi$ for all $t\in\mathbb{R}$, and

  * •

for every $a\in D(\alpha_{-\beta i/2})$ we have that
$\psi(a^{*}a)=\psi(\alpha_{-\beta i/2}(a)\,\alpha_{-\beta i/2}(a)^{*})$.

Note that a _KMS β-state for $\alpha$_ is defined analogously with the
additional criteria of being a state.

###  2.4. Dynamical systems of groupoid algebras

Let $\mathcal{G}$ be an étale groupoid, and
$c:\mathcal{G}\rightarrow\mathbb{R}$ be a continuous groupoid homomorphism. We
get a group homomorphism
$\alpha^{c}:\mathbb{R}\rightarrow\text{Aut}(C^{*}(\mathcal{G}))$ by extending
the following function, defined by

$\alpha_{t}^{c}(f)(x)=e^{itc(x)}\,f(x),$

for $t\in\mathbb{R}$, $f\in C_{c}(\mathcal{G})$ and $x\in\mathcal{G}$. It is
also clear that if $z\in\mathbb{C}$ we have that $\alpha_{z}^{c}(f)\in
C_{c}(\mathcal{G})$ for all $f\in C_{c}(\mathcal{G})$, hence every $f\in
C_{c}(\mathcal{G})$ is analytic for $\alpha^{c}$. When $\mathcal{G}$ is a
principal groupoid Renault is able to classify all the KMSβ-states for
$\alpha^{c}$ in terms of probability measures on the unit space
$\mathcal{G}^{(0)}$ that are _quasi-invariant with Radon-Nikodym cocycle
$e^{-\beta c}$_, c.f. [Renault1980]. A regular Borel measure $\mu$ on the unit
space $\mathcal{G}^{(0)}$ is _quasi-invariant with Radon-Nikodym cocycle
$e^{-\beta c}$_ if the pullbacks $r^{*}\mu$ and $s^{*}\mu$ are equivalent and
$\frac{d(r^{*}\mu)}{d(s^{*}\mu)}=e^{-\beta c}$. In [Renault1980] he defines
the pullback of a measure using Equation 1.1. However, we want to think of the
pullback differently, which we make precise in Section 3. Denote all the
quasi-invariant measures with Radon-Nikodym cocycle $e^{-\beta c}$ by
$\Delta(e^{-\beta c})$.

For non-principal groupoids, Neshveyev is able to generalize Renault’s
description, c.f. [Neshveyev2013]. Combining [Neshveyev2013]*Theorem 1.3 with
his remark that if $\beta\neq 0$ and $\mu\in\Delta(e^{-\beta c})$, then
$\mathcal{G}_{u}^{u}\subset c^{-1}(0)$ for $\mu$-a.e. $u\in\mathcal{G}^{(0)}$,
we get the following result, which we will need in this paper.

######  Proposition 2.1.

[Neshveyev2013]
Let $\mathcal{G}$ be an étale groupoid, $c:\mathcal{G}\rightarrow\mathbb{R}$
be a continuous groupoid homomorphism and
$\beta\in\mathbb{R}\setminus\\{0\\}$. If
$c^{-1}(0)\cap\mathcal{G}_{u}^{u}=\\{u\\}$ for all $u\in\mathcal{G}^{(0)}$ we
have that there is an affine bijection between the set of KMSβ-states for
$\alpha^{c}$ on $C^{*}(\mathcal{G})$ and the set of quasi-invariant
probability measures on $\mathcal{G}^{(0)}$ with Radon-Nikodym cocycle
$e^{-\beta c}$. ∎

######  Remark 2.2.

In [Christensen2023] Christensen is able to further build on Neshveyev’s work
and attains a similar description, only of KMSβ-_weights_ instead of states.
Hence, Proposition 2.1 remains true in the more general case of KMSβ-weights
if we remove the criteria of the measures being probability measures.

###  2.5. Topological graphs

A lot of the theory for $C^{*}$-algebras associated to topological graphs was
first developed by Katsura in [Katsura2003]. Katsura associated a
$C^{*}$-algebra to the quadruple $E=(E^{0},E^{1},r,s)$ where $E^{0}$ is
thought of as the vertex space of the graph, $E^{1}$ is thought of as the edge
space of the graph and $r:E^{1}\rightarrow E^{0}$ and $s:E^{1}\rightarrow
E^{0}$ are maps indicating the direction of each edge, so we may think of $E$
as a generalization of directed graphs. To be precise we define a _topological
graph_ $E$ to be a quadruple $(E^{0},E^{1},r,s)$ where $E^{0}$ and $E^{1}$ are
locally compact Hausdorff spaces, $s:E^{1}\rightarrow E^{0}$ is a local
homeomorphism, called the _source map_ , and $r:E^{1}\rightarrow E^{0}$ is a
continuous map called the _range map_. We call $E^{0}$ and $E^{1}$ the
_vertex_ space and _edge_ space respectively. In the case where both $E^{0}$
and $E^{1}$ are second-countable we will say that the topological graph $E$ is
second-countable.

It was shown by Yeend, c.f. [Yeend2007, Yeend2006], that the $C^{*}$-algebra
constructed by Katsura admits a groupoid model. Yeend’s description is for
higher rank topological graphs, for rank 1 topological graphs there exists
another equivalent description due to Kumijan and Li, c.f. [Kumjian2017]. It
is their description that we will be using.

For paths in topological graphs we use the following convention: if we have
edges $\\{a_{1},a_{2},\dots,a_{n}\\}\subset E^{1}$ such that
$s(a_{k})=r(e_{k+1})$ we get a path of length $n$ by composing:
$a=a_{1}a_{2}\cdots a_{n}$, we denote its length by $|a|=n$. One can similarly
define infinite paths. We then define the set $E^{n}$ to be the set of all
paths $a$ such that $|a|=n$, we similarly define $E^{\infty}$ to be the set of
all paths of infinite length. We can give $E^{n}$ the structure of a locally
compact Hausdorff space by considering it as a subspace of
$\prod_{k=1}^{n}E^{1}$. We can similarly give $E^{\infty}$ the structure of a
Hausdorff space by considering it as a subspace of
$\prod_{k=1}^{\infty}E^{1}$, note that $E^{\infty}$ may not be locally
compact.

We define the _finite path space_ $E^{*}=\bigsqcup_{n=0}^{\infty}E^{n}$ and
endow it with the disjoint union topology making it a locally compact
Hausdorff space. We can extend the range and source maps to all of $E^{*}$ by
defining

$r(a)=r(a_{1}),\ s(a)=s(a_{|a|})$

whenever $|a|\geq 1$. For vertices $v\in E^{0}$ we define $r(v)=v=s(v)$. We
can similarly extend the range map to $E^{\infty}$ as well. Note that the
source of an infinite path is not well-defined.

We would also like to capture the vertices that have edges pointing towards
them, but not too many edges. We call these vertices for _regular_. In the
discrete graph case the regular vertices are defined as the vertices $v$ such
that the inverse image $r^{-1}(v)$ is non-empty and finite. For a topological
graph $E$ we say that a vertex $v\in E^{0}$ is _regular_ if there exists a
relatively compact open neighborhood $U\subset E^{0}$ about $v$ such that the
inverse image $r^{-1}(\overline{U})$ is compact and $r(r^{-1}(U))=U$. We
denote the set of regular vertices as $E^{0}_{\text{reg}}$. We call the
vertices that are not regular for _singular_ and denote them as
$E^{0}_{\text{sng}}=E^{0}\setminus E^{0}_{\text{reg}}$. Finally, we define a
finite path to be _singular_ if the source of the path is a singular vertex.
We denote the set of singular finite paths as $E^{*}_{\text{sng}}=E^{*}\cap
s^{-1}(E^{0}_{\text{sng}})$.

###  2.6. The boundary path space and graph groupoid

By [Kumjian2017]*Proposition 4.6 Yeend’s _boundary path space_ can be
described as the disjoint union

$\partial E=E^{*}_{\text{sng}}\sqcup E^{\infty},$

where it is endowed with a suitable topology. Before we say what the topology
on the boundary path space is it will be useful to introduce some notation.

######  Definition 2.3.

Let $E$ be a topological graph and $a\in E^{*}\sqcup E^{\infty}$. Write
$a=a_{1}a_{2}\cdots a_{|a|}$. For $k,n\in\mathbb{N}$ such that $k\leq
n\leq|a|$ we will write

$\displaystyle a(n)$ $\displaystyle=a_{1}a_{2}\cdots a_{n},$ $\displaystyle
a(k,n)$ $\displaystyle=a_{k}a_{k+1}\cdots a_{n},$ $\displaystyle a(0)$
$\displaystyle=r(a).$

If we want to refer to a specific edge in the path we will simply use the
subscript notation $a_{n}$ to denote the $n$-th edge of $a$.

######  Lemma 2.4.

Let $E$ be a topological graph and $a\in E^{*}\sqcup E^{\infty}$. Then the map
$a\mapsto a(n)$ is continuous for every $0\leq n\leq|a|$.

###### Proof.

If $a\in E^{*}$, suppose $k\geq n$ and let

$p^{k}_{n}:\prod_{i=1}^{k}E^{1}\rightarrow\prod_{i=1}^{n}E^{1}$

be the projection onto the first $n$ entries. By universality of the product
topology, $p^{k}_{n}$ is continuous and restricts to a continuous map

$q^{k}_{n}:E^{k}\rightarrow E^{n}.$

Then the map $a\mapsto a(n)$ is simply

$\bigsqcup_{k=n}^{\infty}q^{k}_{n}:\bigsqcup_{k=n}^{\infty}E^{k}\rightarrow
E^{n},$

which is continuous by universality of the disjoint union topology.

If $a\in E^{\infty}$ the map $a\mapsto a(n)$ is simply the restriction of the
projection

$p_{n}:\prod_{i=1}^{\infty}E^{1}\rightarrow\prod_{i=1}^{n}E^{1}$

to $E^{\infty}$, which is continuous by universality of the product topology.
∎

For subsets $S\subset E^{*}$ we define

$Z(S)=\\{a\in\partial E\ |\ a(n)\in S\text{ for some }n\text{ satisfying
}0\leq n\leq|a|\\}.$

It is these sets that allows us to endow the boundary path space with a
locally compact Hausdorff topology: Let $E$ be a topological graph. The
collection

$\mathcal{B}=\\{Z(U)\setminus Z(K)\ |\ U\text{ is open in }E^{*}\text{ and
}K\text{ is compact in }E^{*}\\}$

form a base for a locally compact Hausdorff topology on the boundary path
space $\partial E$. See for example [Schafhauser2018]*Proposition 3.2 for a
proof of this fact. We want to extract the following result from Schafhauser’s
proof, as it will be useful later.

######  Lemma 2.5.

[Schafhauser2018]
Let $E$ be a topological graph and $K\subset E^{*}$ be compact. Then the set
$Z(K)$ is compact in $\partial E$. ∎

It is clear that if $E$ is a second-countable topological graph, then the
boundary path space with this topology is also second-countable.

To construct a groupoid from the boundary path space we need an action from
the natural numbers, including zero. This is done via the _backwards shift
map_ defined as follows. Let $E$ be a topological graph and $a\in\partial
E\setminus E^{0}$. We define a map $\sigma:\partial E\setminus
E^{0}\rightarrow\partial E$ by

$\displaystyle\sigma(a)$ $\displaystyle=a(2,|a|),\text{ if }|a|\geq 2$
$\displaystyle\sigma(a)$ $\displaystyle=s(a),\text{ if }|a|=1.$

By [Schafhauser2018]*Proposition 3.5, the backwards shift map is a local
homeomorphism.

The graph groupoid is then defined as

$\mathcal{G}_{E}=\\{(a,k-l,b)\in\partial E\times\mathbb{Z}\times\partial E\ |\
k,l\geq 0,\ k\leq|a|,\ l\leq|b|,\ \sigma^{k}(a)=\sigma^{l}(b)\\}.$

For open subsets $U,V\subset\partial E$ and non-negative integers
$k,l\in\mathbb{N}_{0}$ we define the set $Z(U,k,l,V)$ to be

$Z(U,k,l,V)=\\{(a,k-l,b)\in\mathcal{G}_{E}\ |\ a\in U,\ b\in V\\}.$

It is well known that the collection of these sets form a base for a locally
compact Hausdorff étale topology on $\mathcal{G}_{E}$ since, you might also
recognize $\mathcal{G}_{E}$ as the Deaconu-Renault groupoid associated to the
action $\mathbb{N}_{0}\times\partial E\rightarrow\partial E$ sending
$(k,a)\mapsto\sigma^{k}(a)$. It is clear that if $E$ is a second-countable
topological graph then $\mathcal{G}_{E}$ becomes second-countable as well
(hence, by our naming convention, $\mathcal{G}_{E}$ is an étale groupoid
whenever $E$ is a second-countable topological graph), hence we may form the
associated groupoid $C^{*}$-algebra, $C^{*}(E):=C^{*}(\mathcal{G}_{E})$.

It should also be clear that the unit space of $\mathcal{G}_{E}$ is
homeomorphic to the boundary path space $\partial E$, hence we are ultimately
interested in studying measures on $\partial E$.

###  2.7. The gauge action

We get an action from the circle group on the $C^{*}$-algebra of a second-
countable topological graph $E$ by dualizing the map
$\Phi:\mathcal{G}_{E}\rightarrow\mathbb{Z}$ defined by $\Phi(a,n,b)=n$.
Explicitly, for $z\in\mathbb{T}$ we get a continuous *-homomorphism
$\gamma_{z}:C_{c}(\mathcal{G}_{E})\rightarrow C_{c}(\mathcal{G}_{E})$ defined
by

$\gamma_{z}(f)(a,n,b)=z^{n}\,f(a,n,b)$

for $f\in C_{c}(\mathcal{G}_{E})$ and $(a,n,b)\in\mathcal{G}_{E}$. By
continuity this extends to give us a unique group homomorphism
$\gamma:\mathbb{T}\rightarrow\text{Aut}(C^{*}(E))$ defined by
$z\mapsto\gamma_{z}$. We get a $C^{*}$-dynamical system,
$(C^{*}(E),\mathbb{R},\alpha^{\Phi})$, where $\alpha^{\Phi}$ is defined by
precomposing $\gamma$ with the exponential map, i.e.
$\alpha^{\Phi}_{t}=\gamma_{e^{it}}$.

##  3\. Sheaves of measures

Instead of trying to immediately describe the pullback of a _regular_ Borel
measure let us take a step back and only consider Borel measures. Let
$f:X\rightarrow Y$ be a local homeomorphism between topological spaces and
$\mu$ be a Borel measure on $Y$. If $U$ is an open subset of $X$ such that
$f|_{U}$ is injective we have that $U\cong f(U)$, and we want the Borel
subsets of $U$ to have the same measure as their image in $f(U)$. So we define
the pullback $f|_{U}^{*}\mu$ of $\mu$ on $U$ by setting
$f|_{U}^{*}\mu(B)=\mu(f(B))$ for each Borel subset $B$ in $U$. If we let

$\mathcal{U}=\\{U\subset X\ |\ U\text{ is open in }X\text{ and }f|_{U}\text{
is injective}\\}$

we get a family of Borel measures $\\{f|_{U}^{*}\mu\\}_{U\in\mathcal{U}}$. We
would like to be able to glue these measures to obtain a global measure on
$X$, and we would like the resulting measure to be independent of our choice
of open cover of $X$.

Sheaf-theory is precisely the tool that axiomatizes such problems and gives us
a clear road map for things we need to prove in order to define the pullback
of a measure. To our knowledge using sheaf-theory to study measures in this
context is all original work.

######  Definition 3.1.

Let $X$ be a topological space. For each open subset $U\subset X$, let

$\mathcal{M}(U)=\\{\mu:\mathcal{B}(U)\rightarrow[0,\infty]\ |\ \mu\text{ is a
Borel measure}\\},$

where $\mathcal{B}(U)$ denotes the Borel $\sigma$-algebra of $U$. For
inclusion of open subsets $V\subset U$ in $X$, we define the restriction map
$\rho^{U}_{V}:\mathcal{M}(U)\rightarrow\mathcal{M}(V)$ by $\mu|_{V}(B)=\mu(B)$
where $\mu\in\mathcal{M}(U)$ and $B\subset V$ is a Borel subset. This is well-
defined since any Borel subset in $V$ is automatically a Borel subset in $U$,
since $V$ is open in $U$. We denote the collection of this data, namely the
sets $\mathcal{M}(U)$ and the restriction maps
$\rho^{U}_{V}:\mathcal{M}(U)\rightarrow\mathcal{M}(V)$, by $\mathcal{M}$.

######  Lemma 3.2.

Let $X$ be a topological space, then $\mathcal{M}$ is a presheaf of sets on
$X$.

###### Proof.

Fix an open subset $U\subset X$ and let $\mu\in\mathcal{M}(U)$. Clearly
$\mu|_{U}=\mu$, so $\rho^{U}_{U}=id_{\mathcal{M}(U)}$. If we have inclusions
of open subsets $W\subset V\subset U$ in $X$, we have for all Borel subsets
$B\subset W$ that

$(\mu|_{V})|_{W}(B)=\mu|_{V}(B)=\mu(B)=\mu|_{W}(B).$

Hence, $\rho^{V}_{W}\circ\rho^{U}_{V}=\rho^{U}_{W}$. This completes the proof
showing that $\mathcal{M}$ is a contravariant functor from the topology of any
topological space to the category of sets. ∎

For general topological spaces $X$ we won’t get that $\mathcal{M}$ is a sheaf.
To prove the locality axiom we require that any open cover $\mathcal{U}$ of
$X$ has a countable subcover, which isn’t guaranteed. The need for this comes
from the additivity axiom for measures. We do however get a sheaf for second-
countable spaces.

######  Proposition 3.3.

Let $X$ be a second-countable space, then $\mathcal{M}$ is a sheaf of sets on
$X$.

###### Proof.

Note first that second-countability of $X$ implies that any subspace of $X$ is
also second-countable. Hence, it suffices to prove that locality and gluing
holds for $X$ to conclude that they hold for any open subset of $X$.

Let $\mathcal{U}$ be an open cover of $X$. Since $X$ is second-countable we
get a countable sub-cover $\mathcal{V}$ of $\mathcal{U}$. Let
$\mathcal{V}=\\{V_{i}\\}_{i\in\mathbb{N}}$ be an indexing of $\mathcal{V}$.

For subsets $S\subset X$ we introduce the notation
$S(i)=S\cap(V_{i}\setminus(V_{1}\cup...\cup V_{i-1}))$. Clearly
$S=\bigcup_{i\in\mathbb{N}}S(i)$ with $S(i)\cap S(j)=\emptyset$ for $i\neq j$
and each $S(i)\subset V_{i}$.

_Locality:_
Suppose we have Borel measures $\mu,\nu\in\mathcal{M}(X)$ such that
$\mu|_{U}=\nu|_{U}$ for all $U\in\mathcal{U}$. Then for any Borel subset
$B\subset X$ we have that

$\mu(B)=\mu\left(\bigcup_{i\in\mathbb{N}}B(i)\right)=\sum_{i\in\mathbb{N}}\mu(B(i))=\sum_{i\in\mathbb{N}}\mu|_{V_{i}}(B(i))=\sum_{i\in\mathbb{N}}\nu|_{V_{i}}(B(i))=\nu(B).$

_Gluing:_
Suppose we have a Borel measures $\mu_{U}\in\mathcal{M}(U)$ for each
$U\in\mathcal{U}$ such that $\mu_{U}|_{U\cap V}=\mu_{V}|_{U\cap V}$ for all
$U,V\in\mathcal{U}$. Define $\mu_{i}=\mu_{V_{i}}$. We define a global Borel
measure $\mu\in\mathcal{M}(X)$ by

$\mu(B)=\sum_{i\in\mathbb{N}}\mu_{i}(B(i)),$

where $B\subset X$ is a Borel subset. The only non-trivial thing we need to
check to ensure that $\mu$ is a Borel measure is additivity:

$\displaystyle\mu\left(\bigcup_{j\in\mathbb{N}}B_{j}\right)$
$\displaystyle=\sum_{i\in\mathbb{N}}\mu_{i}\left(\bigcup_{j\in\mathbb{N}}B_{j}(i)\right)=\sum_{i\in\mathbb{N}}\sum_{j\in\mathbb{N}}\mu_{i}(B_{j}(i))=\sum_{j\in\mathbb{N}}\sum_{i\in\mathbb{N}}\mu_{i}(B_{j}(i))=\sum_{j\in\mathbb{N}}\mu(B_{j}),$

where $\\{B_{j}\\}_{j\in\mathbb{N}}$ is a collection of mutually disjoint
Borel subsets in $X$. Note that we can interchange the sums in the last line
because we are summing positive numbers.

Now let $U\in\mathcal{U}$ and $B\subset U$ be a Borel subset. To see that
$\mu|_{U}=\mu_{U}$ is a matter of unraveling definitions and noting that each
$B(i)\subset U\cap V_{i}$, hence $\mu_{i}|_{U\cap V_{i}}(B(i))=\mu_{U}|_{U\cap
V_{i}}(B(i))$. ∎

Note that the sets $\mathcal{M}(U)$, _almost_ has the structure of a real
vector space, since we can add measures and scale them by non-negative real
numbers. One could consider the group completion of $\mathcal{M}(U)$ by adding
formal inverses to different measures, i.e. by adding _virtual measures_. If
we do this we get that $\mathcal{M}$ takes values in $\mathbf{Ab}$, so for
second-countable spaces, $\mathcal{M}$ would be a sheaf of abelian groups (in
fact a sheaf of real vector spaces). We do not explore this further in this
paper, but it is worth noting.

We now give an example of a (not second-countable) space for which locality
does not hold.

######  Example 3.4.

Consider $\mathbb{R}$ with the discrete topology. Let
$\mathcal{U}=\\{\\{x\\}\\}_{x\in\mathbb{R}}$ be an open covering of
$\mathbb{R}$. Let $\mu$ be the measure defined by

$\mu(B)=\left\\{\begin{matrix}[l]0&\text{if }B\text{ is countable or
finite},\\\ \infty&\text{if }B\text{ is uncountable}.\end{matrix}\right.$

This clearly defines a measure. It is clear that $\mu$ agrees with the zero
measure on all restrictions to $\\{x\\}\in\mathcal{U}$, however $\mu$ is not
the zero measure, so locality doesn’t hold.

We are however mostly interested in _regular_ Borel measures. Some authors
give different definitions of a regular measure, we will say that a Borel
measure is regular if it is both inner- and outer-regular _and_ finite on
compact sets. Recall that for locally compact Hausdorff spaces we have that a
Borel measure is regular if and only if it is finite on compact sets.

We now want to find the requirements on a topological space for which
$\mathcal{M}_{\text{reg}}$, the family of _regular_ Borel measures, becomes a
sheaf of sets.

######  Lemma 3.5.

Let $X$ be a topological space, then $\mathcal{M}_{\text{reg}}$ is a sub-
presheaf of $\mathcal{M}$ on $X$.

###### Proof.

We clearly have that $\mathcal{M}_{\text{reg}}(U)\subset\mathcal{M}(U)$ for
all open subset $U\subset X$. We want to define the restriction of
$\mathcal{M}_{\text{reg}}$ to be the restriction of $\mathcal{M}$, i.e. that
it is given by the composition

$\textstyle{\mathcal{M}_{\text{reg}}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{M}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{M}(U)}$

for inclusions of open subsets $U\subset V\subset X$. We want to check that
this composition factors through $\mathcal{M}_{\text{reg}}(U)$, then we
immediately get that $\mathcal{M}_{\text{reg}}$ is a sub-presheaf of
$\mathcal{M}$ on $X$. To do this we only need to check that the restriction of
a regular Borel measure (viewed as a section of $\mathcal{M}(V)$) is again a
regular Borel measure. Doing this is standard procedure, so we omit the
details here. ∎

######  Proposition 3.6.

Let $X$ be a second-countable locally compact Hausdorff space. Then
$\mathcal{M}_{\text{reg}}$ is a sub-sheaf of $\mathcal{M}$ on $X$.

###### Proof.

Note that since $\mathcal{M}_{\text{reg}}$ is a sub-presheaf of $\mathcal{M}$
we only need to check that $\mathcal{M}_{\text{reg}}$ is a sheaf of sets on
$X$ to conclude that $\mathcal{M}_{\text{reg}}$ is a sub-sheaf of
$\mathcal{M}$.

We have that any subspace of $X$ is second-countable locally compact and
Hausdorff, hence it suffices to prove that locality and gluing holds for $X$
to conclude that it holds for any open subset of $X$.

Let $\mathcal{U}$ be an open cover of $X$. Locality for
$\mathcal{M}_{\text{reg}}$ follows by the same argument as for $\mathcal{M}$.
To prove gluing, suppose we have regular Borel measures
$\mu_{U}\in\mathcal{M}_{\text{reg}}(U)$ for each $U\in\mathcal{U}$ such that
$\mu_{U}|_{U\cap V}=\mu_{V}|_{U\cap V}$ for all $U,V\in\mathcal{U}$. We get a
global measure $\mu\in\mathcal{M}(X)$ that restricts to each $\mu_{U}$ for
$U\in\mathcal{U}$ by Proposition 3.3. It remains to check that $\mu$ is in
fact regular. Let

$\mathcal{V}=\\{V\subset X\ |\ V\text{ is open in }X\text{ and
}\overline{V}\subset U\text{ for some }U\in\mathcal{U}\\}.$

This is an open cover of $X$: For any $x\in X$ let $U\in\mathcal{U}$ be a
neighborhood about $x$. Then $X\setminus U$ is closed, and since $X$ is
locally compact Hausdorff, $X$ is in particular regular, so we can find
disjoint open subsets $V$ and $W$ containing $x$ and $X\setminus U$
respectively. Then $\overline{V}\subset U$, showing that $V\in\mathcal{V}$.

Let now $K\subset X$ be compact, then $\mathcal{V}$ is an open cover of $K$,
so it admits a finite sub-cover $\\{V_{i}\\}_{i=1}^{n}$ of $K$. Let
$U_{i}\in\mathcal{U}$ be such that $V_{i}\subset\overline{V_{i}}\subset
U_{i}$. Then $K\cap\overline{V_{i}}\subset U_{i}$ is compact. Thus,

$\mu(K)\leq\sum_{i=1}^{n}\mu(K\cap\overline{V_{i}})=\sum_{i=1}^{n}\mu|_{U_{i}}(K\cap\overline{V_{i}})=\sum_{i=1}^{n}\mu_{U_{i}}(K\cap\overline{V_{i}})<\infty.$

Hence, since $X$ is locally compact Hausdorff we get that $\mu$ is in fact
regular, proving that gluing holds for $\mathcal{M}_{\text{reg}}$. ∎

With this in place we can easily define the pullback of a regular Borel
measure.

######  Proposition 3.7.

Let $f:X\rightarrow Y$ be a local homeomorphism between second-countable
locally compact Hausdorff spaces and $\mu$ be a regular Borel measure on $Y$.
Then there exists a unique regular Borel measure $f^{*}\mu$ on $X$, the
_pullback of $\mu$_, such that $f^{*}\mu(U)=\mu(f(U))$ for all open subsets
$U\subset X$ where $f|_{U}$ is injective.

###### Proof.

Let

$\mathcal{U}=\\{U\subset X\ |\ U\text{ is open in }X\text{ and }f|_{U}\text{
is injective}\\}.$

Since $f$ is a local homeomorphism this is an open cover of $X$. For each
$U\in\mathcal{U}$ we get a regular Borel measure
$f^{*}\mu_{U}\in\mathcal{M}_{\text{reg}}(U)$ by the equation
$f^{*}\mu_{U}(B)=\mu(f(B))$, where $B\subset U$ is a Borel subset. The fact
that this defines a Borel measure is obvious. Regularity follows as such: Let
$K\subset U$ be compact, then $f^{*}\mu_{U}(K)=\mu(f(K))<\infty$ by
compactness of $f(K)$ which follows by continuity of $f$.

Now we need to show that these measures agree on intersections. Let
$U,V\in\mathcal{U}$. Then for any Borel subset $B\subset U\cap V$ we have that

$f^{*}\mu_{U}|_{U\cap V}(B)=f^{*}\mu_{U}(B)=\mu(f(B))=f^{*}\mu_{V}|_{U\cap
V}(B).$

Since $\mathcal{M}_{\text{reg}}$ is a sheaf on $X$ by Proposition 3.6, we get
that there exists a unique regular Borel measure $f^{*}\mu$ on $X$ such that
for any $U\in\mathcal{U}$

$f^{*}\mu(U)=f^{*}\mu|_{U}(U)=f^{*}\mu_{U}(U)=\mu(f(U)).\qed$

We are now also able to prove that this definition of the pullback agrees with
the one obtained by Equation 1.1. Even though the pullback as defined through
Equation 1.1 is a well-known construction, we choose to present the
construction here. We will omit the proof of this next result as it is easily
obtained using standard methods.

######  Lemma 3.8.

Let $X$ and $Y$ be topological spaces, $\varphi:X\rightarrow Y$ a local
homeomorphism and $f\in C_{c}(X)$. Then the function

$y\mapsto\sum_{x\in\varphi^{-1}(y)}f(x)$

from $Y$ to the complex numbers $\mathbb{C}$ is well-defined and continuous
with compact support. ∎

With this Lemma we get a linear map $\mathcal{L}:C_{c}(X)\rightarrow C_{c}(Y)$
defined by

$(\mathcal{L}f)(y)=\sum_{x\in\varphi^{-1}(y)}f(x).$

Maps of this kind are called _transfer operators_ , see for example
[Baladi2000] for an introduction to their use in dynamics.

######  Proposition 3.9.

Let $\varphi:X\rightarrow Y$ be a local homeomorphism between locally compact
Hausdorff spaces and $\mu$ be a regular Borel measure on $Y$. Then there
exists a unique regular Borel measure $\varphi^{*}\mu$ on $X$ such that

$\int_{X}fd\varphi^{*}\mu=\int_{Y}\sum_{x\in\varphi^{-1}(y)}f(x)d\mu(y),$

where $f\in C_{c}(X)$.

###### Proof.

The proof follows by Lemma 3.8 along with Riesz representation Theorem. ∎

######  Proposition 3.10.

Let $\varphi:X\rightarrow Y$ be a local homeomorphism between second-countable
locally compact Hausdorff spaces and $\mu$ be a regular Borel measure on $Y$.
Then the pullback of $\mu$ as in Proposition 3.7 equals the one in Proposition
3.9.

###### Proof.

Let $\lambda$ be the pullback of $\mu$ as in Proposition 3.7 and $\nu$ be the
pullback of $\mu$ as in Proposition 3.9. Let

$\mathcal{U}=\\{U\subset X\ |\ U\text{ is open in }X\text{ and
}\varphi|_{U}\text{ is injective}\\}.$

This is an open cover of $X$ since $\varphi$ is a local homeomorphism. By
Proposition 3.6 we know that $\mathcal{M}_{\text{reg}}$ is a sheaf on $X$, so
we need only check that $\lambda|_{U}=\nu|_{U}$ for each $U\in\mathcal{U}$ to
conclude that $\lambda=\nu$. Let $U\in\mathcal{U}$ and $B\subset U$ be a Borel
subset. Then

$\displaystyle\nu|_{U}(B)$ $\displaystyle=\nu(B)$
$\displaystyle=\int_{Y}\sum_{x\in\varphi^{-1}(y)}\chi_{B}(x)d\mu(y)$
$\displaystyle=\int_{Y}\chi_{\varphi(B)}d\mu$
$\displaystyle=\lambda|_{U}(B).\qed$

We end this section by briefly recalling the definition of the _pushforward_
of a Borel measure: Let $X$ and $Y$ be topological spaces, $f:X\rightarrow Y$
be continuous and $\mu$ be a Borel measure on $X$. We get a Borel measure
$f_{*}\mu$ on $Y$, the _pushforward of $\mu$_, defined by
$f_{*}\mu(B)=\mu(f^{-1}(B))$ for Borel subsets $B\subset Y$. It is clear that
for the pushforward of regular measure to be regular it is sufficient for the
function $f$ in this definition to be proper.

##  4\. Quasi-invariant measures on a topological graph

We want to specialize Proposition 2.1 for the graph groupoid,
$\mathcal{G}_{E}$, of a second-countable topological graph. This amounts to
giving a description of the set $\Delta(e^{-\beta\Phi})$ only using data from
the topological graph. Christensen does this for the $\beta=0$ case, i.e. he
obtains a description of the tracial-weights on $C^{*}(E)$, c.f.
[Christensen2022]*Theorem 1.3, note that describing the tracial-weights is a
bit more convoluted than describing the KMSβ-weights for values of $\beta\neq
0$.

The regular Borel measures $\nu$ on $\partial E$ that will be of interest will
turn out to be those that satisfy the following equality:
$\sigma^{*}\nu=e^{\beta}\nu$ on $\partial E\setminus E^{0}$. Since we want to
mainly think about this property we make the following definition.

######  Definition 4.1.

Let $E$ be a second-countable topological graph and $\beta\in\mathbb{R}$. We
say that a regular Borel measure $\nu$ on the boundary path space $\partial E$
is _$\beta$ -quasi-invariant_ if $\sigma^{*}\nu=e^{\beta}\nu$ on $\partial
E\setminus E^{0}$. We denote the set of all $\beta$-quasi invariant measures
by $\mathcal{M}^{\beta}_{\text{quasi}}(E)$.

In the next result the range and source maps, denoted $r$ and $s$
respectively, will refer to the range and source maps of the groupoid
$\mathcal{G}_{E}$, not the topological graph.

######  Lemma 4.2.

Let $E$ be a second-countable topological graph,
$\beta\in\mathbb{R}\setminus\\{0\\}$,
$\Phi:\mathcal{G}_{E}\rightarrow\mathbb{Z}$ be given by $\Phi(a,n,b)=n$. Then
$\Delta(e^{-\beta\Phi})\cong\mathcal{M}^{\beta}_{\text{quasi}}(E)$.

###### Proof.

We first want an easier way to check if a regular Borel measure is quasi-
invariant with Radon-Nikodym cocycle $e^{-\beta\Phi}$. To that end, suppose
that $\nu\in\Delta(e^{-\beta\Phi})$ and let $B\subset\mathcal{G}_{E}$ be a
Borel subset. Then we have that $B\cap\Phi^{-1}(\\{k\\})$ is a Borel subset
for any $k\in\mathbb{Z}$ since $\Phi$ is continuous. Hence,

$\displaystyle r^{*}\nu(B)$ $\displaystyle=\int_{B}e^{-\beta\Phi}d(s^{*}\nu)$
$\displaystyle=\sum_{k\in\mathbb{Z}}\int_{B\cap\Phi^{-1}(\\{k\\})}e^{-\beta
k}d(s^{*}\nu)$ $\displaystyle=\sum_{k\in\mathbb{Z}}e^{-\beta
k}s^{*}\nu(B\cap\Phi^{-1}(\\{k\\})).$

So we have that $\nu\in\Delta(e^{-\beta\Phi})$ if and only if

(4.1) $r^{*}\nu=\sum_{k\in\mathbb{Z}}e^{-\beta
k}(s^{*}\nu)|_{\Phi^{-1}(\\{k\\})},$

where the notation $(s^{*}\nu)|_{\Phi^{-1}(\\{k\\})}$ simply means that

$(s^{*}\nu)|_{\Phi^{-1}(\\{k\\})}(B)=s^{*}\nu(B\cap\Phi^{-1}(\\{k\\}))$

for Borel subsets $B\subset\mathcal{G}_{E}$. Note that if
$W\subset\mathcal{G}_{E}$ is an open bisection we have that Equation 4.1 takes
the form

$\nu(r(W))=\sum_{k\in\mathbb{Z}}e^{-\beta k}\nu(s(W\cap\Phi^{-1}(\\{k\\}))).$

Now consider the collection

$\mathcal{U}=\\{U\subset\partial E\setminus E^{0}\ |\ U\text{ is open in
}\partial E\text{ and }\sigma|_{U}\text{ is injective}\\}.$

Since $\sigma$ is a local homeomorphism we have that $\mathcal{U}$ is an open
cover of $\partial E\setminus E^{0}$. By Proposition 3.6,
$\mathcal{M}_{\text{reg}}$ is a sheaf on $\partial E\setminus E^{0}$, so it
suffices to check that $\sigma^{*}\nu=e^{\beta}\nu$ on each $U\in\mathcal{U}$
to conclude that they are equal on $\partial E\setminus E^{0}$.

We claim that for any $U\in\mathcal{U}$ the basic open set
$Z(U,1,0,\sigma(U))\subset\mathcal{G}_{E}$ is a bisection. Indeed, any element
in $Z(U,1,0,\sigma(U))$ is of the form $(a,1,\sigma(a))$ by injectivity of
$\sigma|_{U}$. Hence, the equations

$\displaystyle a$ $\displaystyle=r(a,1,\sigma(a))=r(b,1,\sigma(b))=b,$
$\displaystyle\sigma(a)$
$\displaystyle=s(a,1,\sigma(a))=s(b,1,\sigma(b))=\sigma(b),$

show that $r|_{Z(U,1,0,\sigma(U))}$ and $s|_{Z(U,1,0,\sigma(U))}$ are
injective.

Fix $U\in\mathcal{U}$ and let $B\subset U$ be a Borel subset. It is clear that
$Z(B,1,0,\sigma(B))$ is a Borel subset of $Z(U,1,0,\sigma(U))$ and that
$Z(B,1,0,\sigma(B))\cap\Phi^{-1}(\\{k\\})$ is nonempty if and only if $k=1$,
and in this case
$Z(B,1,0,\sigma(B))\cap\Phi^{-1}(\\{1\\})=Z(B,1,0,\sigma(B))$. Hence, by
Equation 4.1 we get that

$\displaystyle\nu(B)$ $\displaystyle=r^{*}\nu(Z(B,1,0,\sigma(B)))$
$\displaystyle=e^{-\beta}s^{*}\nu(Z(B,1,0,\sigma(B)))$
$\displaystyle=e^{-\beta}\nu(\sigma(B)).$

So we indeed have that $\sigma^{*}\nu=e^{\beta}\nu$ on each $U\in\mathcal{U}$,
so we have that $\sigma^{*}\nu=e^{\beta}\nu$ on $\partial E\setminus E^{0}$.

For the converse we will show that Equation 4.1 holds. By Proposition 3.6,
$\mathcal{M}_{\text{reg}}$ is a sheaf on $\mathcal{G}_{E}$, so it suffices to
check that Equation 4.1 holds on each open subset in some suitable open cover
of $\mathcal{G}_{E}$. The open cover we will use is the following:

$\mathcal{V}=\\{Z(U,m,n,V)\ |\ m,n\in\mathbb{N}_{0},U\in\mathcal{U}_{m}\text{
and }V\in\mathcal{U}_{n}\\},$

where

$\mathcal{U}_{k}=\\{U\subset\partial E\setminus E^{k-1}\ |\ U\text{ is open in
}\partial E\setminus E^{k-1}\text{ such that }\sigma^{k}|_{U}\text{ is
injective}\\}.$

We first show that $\mathcal{V}$ is an open cover of $\mathcal{G}_{E}$: Let
$(a,m-n,b)\in\mathcal{G}_{E}$. Then $|a|\geq m$ and $|b|\geq n$, so
$a\in\partial E\setminus E^{m-1}$ and $b\in\partial E\setminus E^{n-1}$. For
every $k\in\mathbb{N}_{0}$ we have that $\sigma^{k}$ is a local homeomorphism,
hence there exists a $U\in\mathcal{U}_{m}$ and a $V\in\mathcal{U}_{n}$ such
that $a\in U$ and $b\in V$. Hence, $(a,m-n,b)\in Z(U,m,n,V)$ where
$Z(U,m,n,V)\in\mathcal{V}$.

Next we claim that each $Z(U,m,n,V)\in\mathcal{V}$ is a bisection. Indeed, let
$(a_{1},m-n,b_{1}),(a_{2},m-n,b_{2})\in Z(U,m,n,V)$ and suppose
$r(a_{1},m-n,b_{1})=r(a_{2},m-n,b_{2})$. By definition of the range map we get
that $a_{1}=a_{2}$. Then we get that

$\sigma^{n}(b_{1})=\sigma^{m}(a_{1})=\sigma^{m}(a_{2})=\sigma^{n}(b_{2}).$

By injectivity of $\sigma^{n}|_{V}$ we get that $b_{1}=b_{2}$, hence
$r|_{Z(U,m,n,V)}$ is injective. Injectivity of $s|_{Z(U,m,n,V)}$ is similarly
proven to be true.

We can now show that any regular Borel measure on $\partial E$ that satisfies
$\sigma^{*}\nu=e^{\beta}\nu$ on $\partial E\setminus E^{0}$ must also satisfy
Equation 4.1. Fix $Z(U,m,n,V)\in\mathcal{V}$ and let $B\subset Z(U,m,n,V)$ be
a Borel subset. It is clear that $B\cap\Phi^{-1}(\\{k\\})$ is nonempty if and
only if $k=m-n$, and in this case $B\cap\Phi^{-1}(\\{m-n\\})=B$. Also since
$r|_{B}$ and $s|_{B}$ are injective, we have that $B=Z(r(B),m,n,s(B))$. By
linearity of the pullback we have that $e^{\beta k}\nu=(\sigma^{k})^{*}\nu$ on
$\partial E\setminus E^{k-1}$ for every $k\in\mathbb{N}$, hence

$\displaystyle e^{\beta m}\nu(r(B))$ $\displaystyle=(\sigma^{m})^{*}\nu(r(B))$
$\displaystyle=\nu(\sigma^{m}(r(B)))$ $\displaystyle=\nu(\sigma^{n}(s(B)))$
$\displaystyle=(\sigma^{n})^{*}\nu(s(B))$ $\displaystyle=e^{\beta
n}\nu(r(B)).$

Note that equality two and four follows by injectivity of $\sigma^{m}|_{U}$
and $\sigma^{n}|_{V}$ and equality three also requires the fact that
$Z(U,m,n,V)$ is a bisection. Hence, $\nu(r(B))=e^{-\beta(m-n)}\nu(s(B))$ so
Equation 4.1 is satisfied, completing the proof. ∎

With these results in place we will no longer need to refer to the source and
range maps of the graph groupoid, hence from now on, when we write $r$ and $s$
we will always refer to the range and source maps of the topological graph.

The isotropy subgroups of the graph groupoid has a particularly nice
interpretation: they measure the period of cycles in the graph. Christensen,
in [Christensen2022], introduces the following definition. Let $E$ be a
second-countable topological graph and $a\in\partial E$. The _periodicity
group of $a$_ is

$Per(a)=\\{k-l\in\mathbb{Z}\ |\ k,l\in\mathbb{N}_{0},\ k,l\leq|a|\text{ and
}\sigma^{k}(a)=\sigma^{l}(a)\\}.$

It is clear that this is isomorphic to the isotropy subgroup
$(\mathcal{G}_{E})^{a}_{a}$. It should also be clear that
$\Phi^{-1}(0)\cap(\mathcal{G}_{E})^{a}_{a}=\\{a\\}$ for all $a\in\partial E$,
hence we are in the situation where Proposition 2.1 holds. Furthermore, it
should be clear that $(\mathcal{G}_{E})^{a}_{a}\subset\Phi^{-1}(0)$ if and
only if $Per(a)$ is trivial.

######  Proposition 4.3.

Let $E$ be a second-countable topological graph and
$\beta\in\mathbb{R}\setminus\\{0\\}$. Then there is an affine bijection
between the set of KMSβ-weights(states) for the gauge-action on $C^{*}(E)$ and
(probability measures in) $\mathcal{M}^{\beta}_{\text{quasi}}(E)$. If
$\nu\in\mathcal{M}^{\beta}_{\text{quasi}}(E)$ (is a probability measure) we
get a KMSβ-weight (state) for the gauge-action on $C^{*}(E)$, $\psi_{\nu}$ by
the following equation

$\psi_{\nu}(f)=\int_{\partial E}f(a,0,a)d\nu(a)$

for all $f\in C_{c}(\mathcal{G}_{E})$.

###### Proof.

By the remark that $\Phi^{-1}(0)\cap Per(a)=\\{a\\}$ for all $a\in\partial E$
we have that Proposition 2.1 holds. Hence, by Remark 2.2, there is an affine
bijection between $\Delta(e^{-\beta\Phi})$ and the set of KMSβ-weights for the
gauge-action on $C^{*}(E)$. By Lemma 4.2 we have that
$\Delta(e^{-\beta\Phi})\cong\mathcal{M}^{\beta}_{\text{quasi}}(E)$, completing
the proof. ∎

Since the only measures that give rise to KMSβ-weights (states) for the gauge
action on $C^{*}(E)$ are the ones that measure the boundary paths with trivial
periodicity groups, it would be nice to be able to describe when the
periodicity groups are trivial. To do this we need to study the cycles in the
topological graph.

For a second-countable topological graph $E$ we say that a path $c\in E^{*}$
is a _cycle_ if $|c|\geq 1$ and $r(c)=s(c)$. We denote the set of all cycles
in $E$ by $\Omega E$. Every cycle $c\in\Omega E$ gives rise to a well-defined
infinite path $c^{\infty}=cc\cdots$. We further say that a path $b\in E^{*}$
is an _exit of $c$_ if $s(b)=r(c)$ and $b$ is not of the form $ac$ for any
path $a\in E^{*}$. We denote the set of exits of $c$ by $E^{*}_{c}$.

It is also interesting to study the infinite paths which are _eventually
cyclic_. To be precise we define a path $a\in E^{\infty}$ to be eventually
cyclic if there exists a cycle $c\in\Omega E$ and an exit $b\in E^{*}_{c}$
such that $a=bc^{\infty}$. We denote the set of all eventually cyclic paths in
$E$ by $\mathcal{C}E$.

######  Proposition 4.4.

Let $E$ be a topological graph and $a\in\partial E$. Then $Per(a)\neq\\{0\\}$
if and only if $a\in\mathcal{C}E$.

###### Proof.

Suppose first that $Per(a)\neq\\{0\\}$. Then there exists integers
$k,l\in\mathbb{N}_{0}$ with $k>l$ such that $\sigma^{k}(a)=\sigma^{l}(a)$.
Define $d=\sigma^{l}(a)$, we want to show that the path segment $d(k-l)$ is a
cycle (recall the notation from Definition 2.3). Indeed,

$\sigma^{k-l}(d)=\sigma^{k-l}(\sigma^{l}(a))=\sigma^{k}(a)=\sigma^{l}(a)=d,$

showing that the path segment $d(k-l)$ is a cycle. To avoid cluttered notation
we define $c=d(k-l)$ to be this cycle, and we will also define $b=a(l)$. It is
then clear that we can write $a=bc^{\infty}$, and if $k$ and $l$ where chosen
minimally we get that $b$ is an exit of $c$, hence we may conclude that
$a\in\mathcal{C}E$.

For the converse statement we now assume that $a\in\mathcal{C}E$. By
definition there exists a cycle $c\in\Omega E$ and an exit $b\in E^{*}_{c}$
such that $a=bc^{\infty}$. Set $k=|b|+|c|$ and $l=|b|$. Since any cycle has a
strictly positive length we get that $k>l$. It is clear that
$\sigma^{k}(a)=\sigma^{l}(a)$, hence $k-l\in Per(a)$ showing that
$Per(a)\neq\\{0\\}$. In fact $Per(a)=|c|\mathbb{Z}$. ∎

This result gives an easy proof of the following interesting fact about the
graph groupoid.

######  Corollary 4.5.

Let $E$ be a topological graph. Then the graph groupoid $\mathcal{G}_{E}$ is
principal if and only if $E$ does not contain any cycles.

###### Proof.

This follows by Proposition 4.4 and the observation that the periodicity
groups are isomorphic to the isotropy subgroups of the graph groupoid. ∎

##  5\. Sub-invariant measures on a topological graph

If our goal is to understand KMSβ-weights for the gauge action on the
$C^{*}$-algebra of a second-countable topological graph, Proposition 4.3 might
not be the best tool, since it will in general be quite difficult to get a
nice description of the boundary path space. In this section we show that
there is an affine bijection between the $\beta$-quasi-invariant measures on
the boundary path space and _$\beta$ -sub-invariant measures_ on the vertex
space of a topological graph. These latter mentioned measures will in general
be easier to get a concrete grasp of.

Recall that the range map of a topological graph $E$ naturally extends to give
us a map $r:\partial E\rightarrow E^{0}$. By definition of the topology on
$\partial E$ and Lemma 2.5 it is clear that this map is not only continuous
but also proper. Thus, any regular Borel measure on $\partial E$ can be
pushforwarded to give a regular Borel measure on $E^{0}$. It is therefore of
interest to study the regular Borel measures on the vertex space $E^{0}$ which
correspond to the KMSβ-weights for the gauge-action on $C^{*}(E)$. Towards
that endeavor we make the following definition.

######  Definition 5.1.

Let $E$ be a second-countable topological graph and $\beta\in\mathbb{R}$. We
define the map
$T:\mathcal{M}_{\text{reg}}(E^{0})\rightarrow\mathcal{M}(E^{0})$ by
$T=r_{*}s^{*}$ where $r_{*}$ and $s^{*}$ are the pushforward and pullback
induced by the range and source maps of the topological graph. We say that a
regular Borel measure $\mu$ on the vertex space $E^{0}$ is _$\beta$ -sub-
invariant_ if $T\mu\leq e^{\beta}\mu$ on $E^{0}$ with equality on
$E^{0}_{\text{reg}}$. We denote the set of all $\beta$-sub-invariant measures
by $\mathcal{M}_{\text{sub}}^{\beta}(E)$.

Note that setting $\beta=0$ in this definition recovers similar definitions by
Christensen and Schafhauser used to study tracial-weights and -states, c.f.
[Christensen2022, Schafhauser2018]. In fact, we want to check that some of
their methods, mainly those by Schafhauser, generalizes to the study of KMSβ-
weights.

The operator $T$ can be thought of as flow in the graph. For a very simple
graph,

$\textstyle{\bullet_{u}}$$\textstyle{\bullet_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$

one computes easily that $T\delta_{v}=\delta_{u}$. Or for a slightly more
sophisticated example,

$\textstyle{\bullet_{u}}$$\textstyle{\bullet_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{f}$$\textstyle{\bullet_{w}}$

we have that $T\delta_{v}=\delta_{u}+\delta_{w}$.

######  Lemma 5.2.

Let $E$ be a second-countable topological graph and $\beta\in\mathbb{R}$. The
pushforward
$r_{*}:\mathcal{M}^{\beta}_{\text{quasi}}(E)\rightarrow\mathcal{M}^{\beta}_{\text{sub}}(E)$
induced by the range map $r:\partial E\rightarrow E^{0}$ is well-defined.

###### Proof.

Since the range map is proper by our discussion at the beginning of this
section we get that the pushforward of a $\beta$-quasi-invariant measure $\nu$
is at least regular. Hence, we only need to check that $T(r_{*}\nu)\leq
e^{\beta}(r_{*}\nu)$ on $E^{0}$ with equality on $E^{0}_{\text{reg}}$.

Let $\mathcal{U}=\\{U\subset E^{1}\ |\ U\text{ is open in }E^{1}\text{ and
}s|_{U}\text{ is injective}\\}$. Since the source map is a local
homeomorphism, $\mathcal{U}$ is an open cover of $E^{1}$. Since $E^{1}$ is
second-countable, $\mathcal{U}$ admits a countable sub-cover
$\\{U_{i}\\}_{i\in\mathbb{N}}$. Fix a Borel subset $B\subset E^{0}$. Then

$\displaystyle T(r_{*}\nu)(B)$
$\displaystyle=\sum_{i\in\mathbb{N}}r_{*}\nu(s(r^{-1}(B)\cap(U_{i}\setminus(U_{1}\cup...\cup
U_{i-1}))))$
$\displaystyle=\sum_{i\in\mathbb{N}}\nu(Z(s(r^{-1}(B)\cap(U_{i}\setminus(U_{1}\cup...\cup
U_{i-1})))))$
$\displaystyle=\sum_{i\in\mathbb{N}}\nu(\sigma(Z(r^{-1}(B)\cap(U_{i}\setminus(U_{1}\cup...\cup
U_{i-1})))))$
$\displaystyle=e^{\beta}\sum_{i\in\mathbb{N}}\nu(Z(r^{-1}(B)\cap(U_{i}\setminus(U_{1}\cup...\cup
U_{i-1}))))$ $\displaystyle=e^{\beta}\nu(Z(r^{-1}(B)))$ $\displaystyle\leq
e^{\beta}\nu(Z(B))$ $\displaystyle=e^{\beta}(r_{*}\nu)(B),$

where the inequality in the penultimate line is an equality if $B\subset
E^{0}_{\text{reg}}$. Note that the transition from line three to line four
follows by the definition of $\mathcal{M}^{\beta}_{\text{quasi}}(E)$ and the
equation for the pullback given in Proposition 3.7. Thus,
$r_{*}\nu\in\mathcal{M}^{\beta}_{\text{sub}}(E^{0})$. ∎

To prove that the pushforward is injective we will use the fact that if we
want to check if two measures are equal on a measurable space, it is enough to
check that they are equal on a $\pi$-system that generates the
$\sigma$-algebra and to check that the space can be covered by a countable
increasing sequence of sets from the $\pi$-system, all of finite measure. See
for example [Cohn2013]*Corollary 1.6.4 for a proof of this fact.

The $\pi$-system we will need in this context is the following.

######  Lemma 5.3.

Let $E$ be a second-countable topological graph. Then the collection

(5.1) $\mathcal{C}=\\{Z(V)\subset\partial E\ |\ V\subset E^{n}\text{ is a
Borel subset and }n\in\mathbb{N}_{0}\\}$

is a $\pi$-system that generate the Borel $\sigma$-algebra of $\partial E$.

###### Proof.

Let $U\subset E^{*}$ be open and $K\subset E^{*}$ be compact. It is enough to
show that $Z(U)\setminus Z(K)$ is in the $\sigma$-algebra generated by
$\mathcal{C}$ to conclude that $\mathcal{C}$ generates the Borel
$\sigma$-algebra of $\partial E$. For $n\in\mathbb{N}_{0}$ we define
$U_{n}=U\cap E^{n}$. The following equality shows that $Z(U)$ is in the
$\sigma$-algebra generated by $\mathcal{C}$:
$Z(U)=Z(\bigcup_{n\in\mathbb{N}_{0}}U_{n})=\bigcup_{n\in\mathbb{N}_{0}}Z(U_{n})$.
We can similarly conclude that $Z(K)$ is in the $\sigma$-algebra generated by
$\mathcal{C}$. Hence, $\mathcal{C}$ generates the Borel $\sigma$-algebra of
$\partial E$.

Now we need to show that $\mathcal{C}$ is indeed a $\pi$-system. Let $U\subset
E^{n}$ and $V\subset E^{m}$ be Borel subsets. We may assume that $n\geq m$.
Define $W=\\{a\in U\ |\ a(m)\in V\\}$. By Lemma 2.4 we have that $W$ is a
Borel subset of $E^{n}$ so $Z(W)\in\mathcal{C}$. Clearly $Z(U)\cap Z(V)=Z(W)$
showing that $\mathcal{C}$ is a $\pi$-system. ∎

######  Lemma 5.4.

Let $E$ be a second-countable topological graph and $\beta\in\mathbb{R}$. The
pushforward
$r_{*}:\mathcal{M}^{\beta}_{\text{quasi}}(E)\rightarrow\mathcal{M}^{\beta}_{\text{sub}}(E)$
induced by the range map $r:\partial E\rightarrow E^{0}$ is injective.

###### Proof.

Fix $\nu_{1},\nu_{2}\in\mathcal{M}^{\beta}_{\text{quasi}}(E)$ such that
$r_{*}\nu_{1}=r_{*}\nu_{2}$. We want to check that $\nu_{1}=\nu_{2}$ on the
$\pi$-system $\mathcal{C}$ in the previous Lemma given by Equation 5.1. To
that effort, suppose that $V\subset E^{n}$ is a Borel subset and let

$\mathcal{U}=\\{U\subset E^{n}\ |\ U\text{ is open in }E^{n}\text{ and
}\sigma^{n}|_{U}\text{ is injective}\\}.$

This is an open cover of $V$ since $\sigma^{n}$ is a local homeomorphism,
furthermore since $E^{n}$ is second-countable we have that $\mathcal{U}$
admits a countable subcover, $\\{U_{i}\\}_{i\in\mathbb{N}}$. By partitioning
$V$ with the $U_{i}$’s it is enough to check that
$\nu_{1}(Z(U_{i}))=\nu_{2}(Z(U_{i}))$ for each $i\in\mathbb{N}$ to conclude
that $\nu_{1}(Z(V))=\nu_{2}(Z(V))$. By definition of being $\beta$-quasi-
invariant measures we get that

$\displaystyle\nu_{1}(Z(U_{i}))$ $\displaystyle=e^{-\beta
n}\nu_{1}(\sigma^{n}(Z(U_{i})))$ $\displaystyle=e^{-\beta
n}\nu_{1}(Z(\sigma^{n}(U_{i})))$ $\displaystyle=e^{-\beta
n}(r_{*}\nu_{1})(\sigma^{n}(U_{i}))$ $\displaystyle=e^{-\beta
n}(r_{*}\nu_{2})(\sigma^{n}(U_{i}))$ $\displaystyle=\nu_{2}(Z(U_{i})).$

Hence, $\nu_{1}=\nu_{2}$ on the $\pi$-system $\mathcal{C}$. To conclude that
$\nu_{1}=\nu_{2}$ on the Borel $\sigma$-algebra of $\partial E$ we need to
show that we can cover $\partial E$ with a sequence of increasing sets from
the $\pi$-system $\mathcal{C}$, with each set having finite measure. We can do
this as follows: consider the collection

$\mathcal{V}=\\{V\subset E^{0}\ |\ V\text{ is relatively compact in
}E^{0}\\}.$

Since $E^{0}$ is locally compact Hausdorff this is an open cover of $E^{0}$,
and since $E^{0}$ is second-countable $\mathcal{V}$ admits a countable
subcover $\\{V_{i}\\}_{i\in\mathbb{N}}$. Define
$W_{i}=\bigcup_{j=1}^{i}V_{j}$. Then each $W_{i}$ is relatively compact in
$E^{0}$ and $W_{i}\subset W_{i+1}$, so we have that the sequence
$\\{Z(W_{i})\\}_{i\in\mathbb{N}}$ is an increasing sequence such that
$\bigcup_{i\in\mathbb{N}}Z(W_{i})=Z\left(\bigcup_{i\in\mathbb{N}}W_{i}\right)=Z(E^{0})=\partial
E$. Finally, we have that
$\nu_{1}(Z(W_{i}))\leq\nu_{1}(Z(\overline{W_{i}}))<\infty$ since $\nu_{1}$ is
a regular measure and $Z(\overline{W_{i}})$ is compact by Lemma 2.5. Hence,
$\nu_{1}=\nu_{2}$, which proves that $r_{*}$ is injective. ∎

Proving surjectivity of the pushforward is not as straight forward as proving
injectivity. The $\beta=0$ case is done by Schafhauser, c.f.
[Schafhauser2018]*Proposition 4.4, and his proof generalizes to all values of
$\beta$ quite easily. To use his method we need to define the _$n$ -th
boundary path space_ $\partial E_{n}$, which for a topological graph $E$ is
defined to be the following disjoint union: $\partial
E_{n}=E^{0}_{\text{sng}}\sqcup\dots\sqcup E^{n-1}_{\text{sng}}\sqcup E^{n}$.
For subsets $S\subset E^{0}\sqcup...\sqcup E^{n}$ we define the set
$Z_{n}(S)=\\{a\in\partial E_{n}\ |\ a(k)\in S\text{ for some }k\text{
satisfying }0\leq k\leq|a|\\}$. The $n$-th boundary path space becomes a
locally compact Hausdorff space with the topology generated by sets
$Z_{n}(U)\setminus Z_{n}(K)$ where $U\subset E^{0}\sqcup...\sqcup E^{n}$ is an
open subset and $K\subset E^{0}\sqcup...\sqcup E^{n}$ is compact. The proof of
this is completely the same as the one for $\partial E$. Of course if $E$ is
second-countable, $\partial E_{n}$ becomes second-countable as well. Note that
for $n=0$ we have that $\partial E_{0}=E^{0}$.

For $n\geq 1$ we define maps $\rho_{n}:\partial E_{n}\rightarrow\partial
E_{n-1}$ by

$\rho_{n}(a)=\left\\{\begin{matrix}[l]a(n-1),&|a|=n,\\\
a,&|a|<n,\end{matrix}\right.$

as well as maps $\rho_{n,\infty}:\partial E\rightarrow\partial E_{n}$ by

$\rho_{n,\infty}(a)=\left\\{\begin{matrix}[l]a(n),&|a|>n,\\\ a,&|a|\leq
n.\end{matrix}\right.$

Note that $\rho_{0,\infty}$ is the range map $r:\partial E\rightarrow E^{0}$.
We may also apply the backwards shift map to the $n$-th boundary path space,
and it will still be a local homeomorphism in this case.

######  Lemma 5.5.

[Schafhauser2018]*Proposition 3.3
Let $E$ be a topological graph. Then the maps $\rho_{n}$ and $\rho_{n,\infty}$
are continuous, proper and surjective. We further have that the maps
$\rho_{n,\infty}$ induces a homeomorphism $\partial E\rightarrow\lim(\partial
E_{n},\rho_{n})$. ∎

######  Lemma 5.6.

Let $E$ be a second-countable topological graph and $\beta\in\mathbb{R}$. The
pushforward
$r_{*}:\mathcal{M}^{\beta}_{\text{quasi}}(E)\rightarrow\mathcal{M}^{\beta}_{\text{sub}}(E)$
induced by the range map $r:\partial E\rightarrow E^{0}$ is surjective.

###### Proof.

Let $\mu\in\mathcal{M}^{\beta}_{\text{sub}}(E^{0})$. We want to define regular
Borel measures $\mu_{n}$ on $\partial E_{n}$ for each $n\in\mathbb{N}_{0}$
such that $(\rho_{n})_{*}\mu_{n}=\mu_{n-1}$. We will then use universality of
the limit to show that there exists a unique regular Borel measure $\nu$ on
$\partial E$ such that $r_{*}\nu=\mu$.

First we set $\mu_{0}=\mu$ as $\partial E_{0}=E^{0}$. We recursively define
$\mu_{n}$ by the equation

$\mu_{n}=e^{-\beta}\sigma^{*}\mu_{n-1}+\mu|_{E^{0}_{\text{sng}}}-e^{-\beta}(T\mu)|_{E^{0}_{\text{sng}}}.$

For Borel subsets $B\subset\partial E_{n}$ we read this equation as follows:

(5.2) $\mu_{n}(B)=e^{-\beta}(\sigma^{*}\mu_{n-1})(B\setminus E^{0})+\mu(B\cap
E^{0}_{\text{sng}})-e^{-\beta}(T\mu)(B\cap E^{0}_{\text{sng}}).$

Since the backwards shift map is a local homeomorphism we get that the
pullback $\sigma^{*}\mu_{n-1}$ becomes a regular Borel measure on $\partial
E_{n}\setminus E^{0}$. Furthermore, since
$\mu\in\mathcal{M}^{\beta}_{\text{sub}}(E^{0})$ we have that
$\mu-e^{-\beta}(T\mu)$ is a regular Borel measure by the inequality $T\mu\leq
e^{\beta}\mu$. Hence, $\mu_{n}$ is a regular Borel measure.

We will prove by induction that $(\rho_{n})_{*}\mu_{n}=\mu_{n-1}$. For $n=1$
we have that $\rho_{1}:\partial E_{1}\rightarrow E^{0}$ is the map $r\sqcup
id_{E^{0}}:E^{1}\sqcup E^{0}_{\text{sng}}\rightarrow E^{0}$ and
$\sigma:\partial E_{1}\setminus E^{0}\rightarrow E^{0}$ is the source map
$s:E^{1}\rightarrow E^{0}$. Hence,

$\displaystyle(\rho_{1})_{*}\mu_{1}$
$\displaystyle=e^{-\beta}(r_{*}s^{*}\mu)+\mu|_{E^{0}_{\text{sng}}}-e^{-\beta}(T\mu)|_{E^{0}_{\text{sng}}}$
$\displaystyle=e^{-\beta}(T\mu)+\mu|_{E^{0}_{\text{sng}}}-e^{-\beta}(T\mu)|_{E^{0}_{\text{sng}}}$
$\displaystyle=e^{-\beta}(T\mu)|_{E^{0}_{\text{reg}}}+\mu|_{E^{0}_{\text{sng}}}$
$\displaystyle=\mu|_{E^{0}_{\text{reg}}}+\mu|_{E^{0}_{\text{sng}}}$
$\displaystyle=\mu$ $\displaystyle=\mu_{0}.$

So assume that $(\rho_{k})_{*}\mu_{k}=\mu_{k-1}$ for all $k\leq n$. To show
that $(\rho_{n+1})_{*}\mu_{n+1}=\mu_{n}$ we will consider the collection
$\mathcal{C}_{n}=\\{Z_{n}(V)\subset\partial E^{n}\ |\ V\subset E^{k}\text{ is
a Borel subset and }0\leq k\leq n\\}$. Just as in Lemma 5.3 this is a
$\pi$-system that generates the Borel $\sigma$-algebra of $\partial E_{n}$.
The case with $V\subset E^{0}$ is special, so we will tackle this last.

Fix a Borel subset $V\subset E^{k}$ where $1\leq k\leq n$. Note that for the
set $Z_{n}(V)$ Equation 5.2 simplifies to

$\mu_{n}(Z_{n}(V))=e^{-\beta}(\sigma^{*}\mu_{n-1})(Z_{n}(V)).$

Consider the family $\mathcal{U}_{k}=\\{U\subset E^{k}\ |\ U\text{ is open in
}E^{k}\text{ and }\sigma|_{U}\text{ is injective}\\}$. Since $\sigma$ is a
local homeomorphism on $E^{k}$ we get that $\mathcal{U}_{k}$ is an open cover
of $E^{k}$, in particular it is an open cover of $V$. Since $E^{k}$ is second-
countable we get that $\mathcal{U}_{k}$ admits a countable subcover of $V$,
denote it by $\\{U_{i}\\}_{i\in\mathbb{N}}$. By partitioning $V$ we get that
$(\rho_{n+1})_{*}\mu_{n+1}(Z_{n}(V))=\mu_{n}(Z_{n}(V))$ if
$(\rho_{n+1})_{*}\mu_{n+1}(Z_{n}(U_{i}))=\mu_{n}(Z_{n}(U_{i}))$ for all
$i\in\mathbb{N}$. So we compute:

$\displaystyle(\rho_{n+1})_{*}\mu_{n+1}(Z_{n}(U_{i}))$
$\displaystyle=\mu_{n+1}(\rho_{n+1}^{-1}(Z_{n}(U_{i})))$
$\displaystyle=\mu_{n+1}((Z_{n+1}(U_{i})))$
$\displaystyle=e^{-\beta}(\sigma^{*}\mu_{n})(Z_{n+1}(U_{i}))$
$\displaystyle=e^{-\beta}\mu_{n}(\sigma(Z_{n+1}(U_{i})))$
$\displaystyle=e^{-\beta}\mu_{n}(Z_{n}(\sigma(U_{i})))$
$\displaystyle=e^{-\beta}\mu_{n}(\rho_{n}^{-1}(Z_{n-1}(\sigma(U_{i}))))$
$\displaystyle=e^{-\beta}(\rho_{n})_{*}\mu_{n}(Z_{n-1}(\sigma(U_{i})))$
$\displaystyle=e^{-\beta}\mu_{n-1}(Z_{n-1}(\sigma(U_{i})))$
$\displaystyle=e^{-\beta}\mu_{n-1}(\sigma(Z_{n}(U_{i})))$
$\displaystyle=e^{-\beta}(\sigma^{*}\mu_{n-1})(Z_{n}(U_{i}))$
$\displaystyle=\mu_{n}(Z_{n}(U_{i})).$

This calculation might seem tedious, but it is nice to exemplify _how_ these
types of calculations are done, as they aren’t the easiest to manoeuvre.

When $V\subset E^{0}$ we have that $Z_{n}(V)\setminus E^{0}=Z_{n}(r^{-1}(V))$.
By partitioning $r^{-1}(V)$ with sets $U\in\mathcal{U}_{1}$ we may assume that
$\sigma|_{r^{-1}(V)}$ is injective. The same calculation as above can now be
carried out, where we also have to drag along the terms involving
$\mu|_{E^{0}_{\text{sng}}}$ and $(T\mu)|_{E^{0}_{\text{sng}}}$. So we have
that $(\rho_{n+1})_{*}\mu_{n+1}=\mu_{n}$ on the $\pi$-system
$\mathcal{C}_{n}$. Making the same argument as we did in Lemma 5.4, we get
that $(\rho_{n+1})_{*}\mu_{n+1}=\mu_{n}$ on $\partial E_{n}$.

The rest of the proof follows in exactly the same manner as in
[Schafhauser2018]*Proposition 4.4. ∎

######  Proposition 5.7.

Let $E$ be a second-countable topological graph and $\beta\in\mathbb{R}$. The
pushforward
$r_{*}:\mathcal{M}^{\beta}_{\text{quasi}}(E)\rightarrow\mathcal{M}^{\beta}_{\text{sub}}(E)$
induced by the range map $r:\partial E\rightarrow E^{0}$ is an affine
bijection.

###### Proof.

The fact that the pushforward is bijective is the combination of Lemma 5.4 and
Lemma 5.6. Linearity of the pushforward gives us that this is an affine
bijection. ∎

We are now able to state and prove the main result of this paper.

######  Theorem 5.8.

Let $E$ be a second-countable topological graph,
$\beta\in\mathbb{R}\setminus\\{0\\}$. There is an affine bijection between the
following three sets: KMSβ-weights (states) for the gauge-action on
$C^{*}(E)$, (probability measures in) $\mathcal{M}^{\beta}_{\text{quasi}}(E)$
and (probability measures in) $\mathcal{M}^{\beta}_{\text{sub}}(E)$.

###### Proof.

The first bijection follows by Proposition 4.3 and the second bijection
follows by Proposition 5.7. For the KMSβ-_states_ case, we only need to check
that the pushforward of a probability measure
$\nu\in\mathcal{M}^{\beta}_{\text{quasi}}(E)$ is a probability measure.
Indeed, by surjectivity of the range map, we get that

$r_{*}\nu(E^{0})=\nu(r^{-1}(E^{0}))=\nu(\partial E)=1.\qed$

It is worth noting that in the case where $E$ is a countable discrete graph,
Theorem 5.8 specializes to a Theorem by Thomsen, c.f. [Thomsen2014]*Theorem
4.6.

######  Example 5.9.

Consider the doubly infinite discrete graph

$\textstyle{\cdots}$$\textstyle{\bullet_{v_{-2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{-2}}$$\textstyle{\bullet_{v_{-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{-1}}$$\textstyle{\bullet_{v_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{0}}$$\textstyle{\bullet_{v_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{1}}$$\textstyle{\bullet_{v_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{3}}$

Any regular Borel measure on $E^{0}=\\{v_{n}\\}_{n\in\mathbb{Z}}$ is of the
form

$\sum_{n\in\mathbb{Z}}a_{n}\delta_{v_{n}}$

with $0\leq a_{n}<\infty$. Note also that $E^{0}=E^{0}_{\text{reg}}$. Thus,
for $\beta\in\mathbb{R}$ we are interested in the measures $\mu$ on $E^{0}$
such that $T\mu=e^{\beta}\mu$. It should be clear that
$T\delta_{v_{n}}=\delta_{v_{n-1}}$, hence we are interested in the measures on
$E^{0}$ such that

$\sum_{n\in\mathbb{Z}}a_{n}\delta_{v_{n-1}}=\sum_{n\in\mathbb{Z}}a_{n}e^{\beta}\delta_{v_{n}}.$

This is the case if and only if $a_{n}=a_{n-1}e^{\beta}$. Thus,

$\mathcal{M}_{\text{sub}}^{\beta}(E)=\left\\{a\sum_{n\in\mathbb{Z}}e^{n\beta}\delta_{v_{n}}\
|\ 0\leq a<\infty\right\\}.$

It should be clear that there are no values of $a$ or $\beta$ that make

$a\sum_{n\in\mathbb{Z}}e^{n\beta}\delta_{v_{n}}$

a probability measure. Hence, there are no KMSβ-states for the gauge action on
the $C^{*}$-algebra of this graph, but there is this one-dimensional ray of
KMSβ-weights for the gauge action for any value of $\beta\neq 0$, by Theorem
5.8.

Note that the conclusion in the following example was reached by Thomsen using
different methods in the case where $Z$ is a compact metric space, c.f.
[Thomsen2011]*Example 6.14.

######  Example 5.10.

Let $Z$ be a second-countable locally compact Hausdorff space, and suppose
that $h:Z\rightarrow Z$ is a surjective local homeomorphism such that
$\\#h^{-1}(z)=n$ for all $z\in Z$. Note that this identifies $Z$ as an
$n$-covering space of $Z$. We get a second-countable topological graph by
setting the vertex and edge spaces equal to $Z$, the range map is set to be
the identity on $Z$ and the source map is set to $h$, i.e. we consider the
topological graph $E=(Z,Z,id_{Z},h)$. In this case we have that

$\mathcal{M}_{\text{sub}}^{\beta}(E)=\\{\mu\in\mathcal{M}_{\text{reg}}(Z)\ |\
h^{*}\mu=e^{\beta}\mu\\}.$

We will show that there can only exist KMSβ-states for the gauge-action on the
associated graph $C^{*}$-algebra if $\beta=\ln(n)$. By Theorem 5.8 this is
equivalent to showing that the pullback
$h^{*}:\mathcal{M}_{\text{reg}}(Z)\rightarrow\mathcal{M}_{\text{reg}}(Z)$ only
has one possible eigenvalue, namely $n$, when viewed as a linear map between
real vector spaces (after taking the group completion of
$\mathcal{M}_{\text{reg}}(Z)$).

Since $Z$ is an $n$-covering space of $Z$ we get that for every $z\in Z$ there
exists an open neighborhood $U_{z}$ about $z$ such that
$h^{-1}(U_{z})=\bigcup_{i=1}^{n}V_{z,i}$ with the $V_{z,i}$’s being mutually
disjoint and $h|_{V_{z,i}}$ being injective. Then
$\mathcal{U}=\\{U_{z}\\}_{z\in Z}$, with each $U_{z}$ as above, is an open
cover of $Z$. By second-countability of $Z$ we have that $\mathcal{U}$ admits
a countable subcover, denote it by $\\{U_{k}\\}_{k\in\mathbb{N}}$. For each
$k\in\mathbb{N}$ let $V_{k,1},\dots,V_{k,n}$ be mutually disjoint open subsets
of $Z$ such that $h^{-1}(U_{k})=\bigcup_{i=1}^{n}V_{k,i}$ and $h|_{V_{k,i}}$
is injective for each $1\leq i\leq n$. Let us now consider the collection

$\mathcal{V}=\\{V_{k,1},\dots,V_{k,n}\\}_{k\in\mathbb{N}}.$

_Claim 1:_ This is an open cover of $Z$. Indeed, let $z\in Z$ and choose
$k\in\mathbb{N}$ such that $h(z)\in U_{k}$. Then $z\in
h^{-1}(U_{k})=\bigcup_{i=1}^{n}V_{k,i}$, hence $z\in V_{k,i}$ for some $1\leq
i\leq n$.

_Claim 2:_ $h\left(\bigcup_{k\in\mathbb{N}}V_{k,j}\right)=Z$ for any $1\leq
j\leq n$. Indeed, let $z\in Z$ and choose $k\in\mathbb{N}$ such that $z\in
U_{k}$. Then, $h^{-1}(U_{k})=\bigcup_{i=1}^{n}V_{k,i}$ and, in particular,
$h|_{V_{k,j}}:V_{k,j}\rightarrow U_{k}$ is a homeomorphism. Thus, there exists
a $w\in V_{k,j}$ such that $h(w)=z$, hence $z\in
h\left(\bigcup_{k\in\mathbb{N}}V_{k,j}\right)$.

With these claims in place, suppose we have a probability measure
$\mu\in\mathcal{M}_{\text{reg}}(Z)$ satisfying $h^{*}\mu=e^{\beta}\mu$.
Partition $Z$ by $\mathcal{V}$, i.e. write $Z$ as a union of mutually disjoint
sets $Z_{k,i}$ where each $Z_{k,i}\subset V_{k,i}$ and
$\bigcup_{k\in\mathbb{N}}Z_{k,i}=\bigcup_{k\in\mathbb{N}}V_{k,i}$. By
Proposition 3.7 we have that

$h^{*}\mu(Z)=\sum_{i=1}^{n}\sum_{k\in\mathbb{N}}\mu(h(Z_{k,i}))=\sum_{i=1}^{n}\mu\left(h\left(\bigcup_{k\in\mathbb{N}}Z_{k,i}\right)\right)=\sum_{i=1}^{n}\mu(Z)=n\mu(Z)=e^{\beta}\mu(Z).$

Since $\mu(Z)=1$ we get that $\beta=\ln(n)$.

It is worth noting that situations similar to this example has been studied
before in many contexts. For example, Exel and Lopes in [Exel2004] study the
situation where $T:X\rightarrow X$ is a local homeomorphism on a compact
metric space. Afsar, an Huef and Raeburn, in [Afsar2014] studied the situation
where $h:Z\rightarrow Z$ is a surjective local homeomorphism on a compact
Hausdorff space.

### Final remarks

Theorem 5.8 attains a description of the KMSβ-weights for the gauge-action on
the $C^{*}$-algebra of a second-countable topological graph. It does not
however, get us any closer to the existence problem of these KMSβ-weights. A
possible direction of further investigation would be to attempt to get a
description of the set $\mathcal{M}^{\beta}_{\text{sub}}(E)$, in complete
generality (ambitious) or for more concrete classes of second-countable
topological graphs.

## References