# Handling Mobility in Low-Power Wide-Area Network

Dali Ismail

Abusayeed Saifullah

Department of Computer Science
Wayne State University
Detroit, Michigan, USA  Department of Computer Science
Wayne State University
Detroit, Michigan, USA

###### Abstract

Despite the proliferation of mobile devices in various wide-area Internet of
Things applications (e.g., smart city, smart farming), current Low-Power Wide-
Area Networks (LPWANs) are not designed to effectively support mobile nodes.
In this paper, we propose to handle mobility in SNOW (Sensor Network Over
White spaces), an LPWAN that operates in the TV white spaces. SNOW supports
massive concurrent communication between a base station (BS) and numerous low-
power nodes through a distributed implementation of OFDM. In SNOW, inter-
carrier interference (ICI) is more pronounced under mobility due to its OFDM
based design. Geospatial variation of white spaces also raises challenges in
both intra- and inter-network mobility as the low-power nodes are not equipped
to determine white spaces. To handle mobility impacts on ICI, we propose a
dynamic carrier frequency offset estimation and compensation technique which
takes into account Doppler shifts without requiring to know the speed of the
nodes. We also propose to circumvent the mobility impacts on geospatial
variation of white space through a mobility-aware spectrum assignment to
nodes. To enable mobility of the nodes across different SNOWs, we propose an
efficient handoff management through a fast and energy-efficient BS discovery
and quick association with the BS by combining time and frequency domain
energy-sensing. Experiments through SNOW deployments in a large metropolitan
city and indoors show that our proposed approaches enable mobility across
multiple different SNOWs and provide robustness in terms of reliability,
latency, and energy consumption under mobility.

##  1 Introduction

Low-Power Wide-Area Network (LPWAN) is an enabling technology for wide-area
Internet-of-Things (IoT) applications such as smart city, agricultural IoT,
and industrial IoT offering long-range (several miles), low-power, and low-
cost communication [22]. With the fast growth of IoT, multiple LPWAN
technologies have emerged recently such as LoRa [24, 13], SigFox [46], IQRF
[20], RPMA [19], DASH7 [9], Weightless-N/P [53], Telensa [1] in the ISM band,
and EC-GSM-IoT [17], NB-IoT [33], and LTE Cat M1 [26, 25] in the licensed
cellular band. To avoid the crowd in the limited ISM band and the cost of
licensed band, SNOW (Sensor Network Over White spaces) is an LPWAN
architecture to support scalable wide-area IoT over the TV white spaces [44,
42, 43]. White spaces are the allocated but locally unused TV spectrum (54 -
698 MHz in the US) [34, 35]. They usually have wide and less crowded spectrum
in rural and most urban areas, with an abundance in rural areas [3].

With a wide range of supported applications, IoT is integrating more mobile
nodes/devices in different domains (e.g. agriculture [51, 52], connected
vehicle [27], healthcare [21], smart city [55]). For example, in agricultural
IoT, the use of drones and tractors is rapidly increasing [29, 14, 15, 52]. It
is expected that by the year 2050, there will be more than 3 billion wearable
sensors [36]. The cellular-based LPWANs rely on wired infrastructure to handle
mobility. Such infrastructure is often not available in rural and remote areas
(e.g., farms, oil fields, etc.). In others, mobility introduces challenges
that are not well-addressed yet. Study on LoRa shows that its performance is
susceptible even to minor human mobility [36, 37].

In this paper, we propose to handle mobility in LPWAN in the white spaces (in
the US) considering SNOW. With the rapid growth of IoT, LPWANs will suffer
from crowded spectrum due to long range, making it critical to exploit white
spaces. SNOW is a highly scalable LPWAN over the white spaces which enables
massive concurrent communication between a base station (BS) and numerous low-
power nodes. It is available as an open-course implementation [47]. Its
physical layer is designed based on a Distributed implementation of OFDM
(orthogonal frequency division multiplexing) for multi-user access, called
D-OFDM. The BS operates on a wide band spectrum which is split into many
orthogonal narrowband subcarriers. A node (non-BS) transmits and receives on a
subcarrier. In SNOW, inter-carrier interference (ICI) is more pronounced under
mobility due to its OFDM based design. Geospatial variation of white spaces
also raises challenges in both intra- and inter-network mobility as the low-
power nodes are not equipped to determine white space. For example, to enable
mobility across different SNOWs, it is challenging for a node to scan the wide
spectrum of the TV band to discover a new BS. Besides, different BSs may be
using different subcarrier widths, which may result in subcarrier misalignment
between the mobile node and the new BS.

In this paper, we address the challenges mentioned above to handle mobility in
SNOW. Specifically, we make the following new contributions.

  * •

To handle mobility impacts on ICI, we propose a dynamic CFO (Carrier Frequency
Offset ) estimation and compensation technique for SNOW which takes into
account Doppler shifts under non-uniform speeds without requiring to know the
speed of the nodes. To circumvent the mobility impacts on geospatial variation
of white space within the same SNOW, we propose a mobility-aware subcarrier
assignment to the nodes.

  * •

To handle inter-SNOW (inter-network) mobility, we propose an energy-efficient
and fast BS discovery technique that considers the trade-off between discovery
latency and energy consumption to allow efficient handoff management. Our
approach utilizes the spectrum information by combining the received signal
features to distinguish between primary users (TV stations) and a SNOW BS. We
also propose a lightweight cross-layer technique feasible at the energy-
constrained SNOW nodes to handle subcarrier alignment by combining time and
frequency domain energy-sensing.

  * •

We implement our proposed mobility handling techniques on SNOW devices and
perform experiments by deploying SNOW in two environments - a large
metropolitan city and an indoor testbed. The experimental results show that
our approaches enable mobility across multiple different SNOWs. The results
also show an improvement of reliability from 80% to 96.6% when our dynamic CFO
estimation and compensation is incorporated.

The rest of the paper is organized as follows. Section 2 overviews related
work. Section 3 presents an overview of SNOW. Section 4 describes the system
model. Section 5 presents our mobility approach. Section 6 presents the
experiments. Section 7 concludes the paper.

##  2 Related Work

Many studies focused on handling mobility in Wireless Sensor Networks (WSNs)
[38, 2, 23, 54, 32, 11] (more can be found in survey [10, 12]) and ad hoc
network [5, 18]. In WSN or WiFi networks, a client has to scan only a
limited/fixed number of channels to discover a new BS. However, those
approaches are not directly applicable to LPWAN. To handle mobility across
networks, cellular LPWANs rely on wired infrastructure. Non-cellular LPWANs
are not yet handling it well. Real experiments show that their performance is
susceptible even to minor human mobility [36]. Spectrum mobility studied in
[56, 48] for cognitive networks enables secondary users to change the
operating frequencies, and is different from device mobility. Device mobility
was studied in [30] for white space network, where every device primarily
relies on the database to determine the white spaces. A mobile device adds a
protection range of $\delta d$ so that any channel blocked within distance
$\delta d$ of current location is not used. Note that this approach does not
work for SNOW as the nodes have no direct access to the Internet (and
database). It first has to discover a BS, associate with it, and rely on it
for spectrum access. Additionally, there has been much work on channel
rendezvous in cognitive radio [57, 45, 4, 8]. Due to technology-specific
nature of SNOW, these techniques cannot be applied to a SNOW.

_Senseless_ [31] is an infrastructure based white space network system where
the devices do not rely on sensing to determine the availability of white
space. They use geo-location service to calculate white space availability at
any location. _Senseless_ then disseminates the availability information to
each device in the network. To address the mobility challenges in white space,
Senseless suggests that every device adds a protection range to determine the
availability of white spaces while it is mobile. This could lead to a huge
spectrum waste depending on the size of the protection area. SNOW differ from
Senseless in that it considers infrastructure-less network system where BSs
are not connected. Besides, the D-OFDM based design requires a different
approach for mobility in SNOW. Also, we incorporate an energy-efficient
sensing approach along with the geo-location service to efficiently handle
nodes mobility. To date, inter-network mobility for non-cellular LPWAN remains
mostly unexplored and it was never studied for SNOW.

##  3 A Brief Overview of SNOW

Figure 1: The SNOW architecture. Figure 2: SNOW hardware.

Here we provide a brief overview of the SNOW architecture [42, 43, 44]. Due to
long transmission (Tx) range (several miles at 0dBm), the nodes in SNOW are
directly connected to the BS, forming a star topology as shown in Fig. 1. We
use ‘node’ to indicate a sensor node. The BS periodically determines white
spaces by providing locations of its own and of all other nodes in a cloud-
hosted database through the Internet. It uses wide white space spectrum as a
single wide channel that is split into narrowband orthogonal subcarriers, each
of equal spectrum width (bandwidth). Each node has a single half-duplex
narrowband radio. It sends/receives on a subcarrier. The nodes are power-
constrained, and do not do spectrum sensing or cloud access. As shown in Fig.
1, the BS uses two radios operating on the same spectrum – one for only
transmission (called Tx radio) and the other for only reception (called Rx
radio) – to facilitate concurrent bidirectional communication.

The physical layer (PHY) of SNOW is designed based on a Distributed
implementation of OFDM for multi-user access, called D-OFDM. D-OFDM splits a
wide spectrum into numerous narrowband orthogonal subcarriers enabling
parallel data streams to/from numerous distributed nodes from/to the BS. A
subcarrier bandwidth is in kHz (e.g., 50kHz, 100kHz, 200kHz, or so depending
on packet size and needed bit rate). The nodes transmit/receive on orthogonal
subcarriers, each using one. A subcarrier is modulated using Binary Phase
Shift Keying (BPSK) or Amplitude Shift Keying (ASK). If the BS spectrum is
split into $m$ subcarriers, it can receive from $m$ nodes simultaneously using
a single antenna. Similarly, it can transmit different data on different
subcarriers through a single transmission. Currently, the sensor nodes in SNOW
use a very simple and lightweight CSMA/CA based MAC (media access control)
protocol like the one used in TinyOS [50].

SNOW was implemented on two hardware platforms [39] – USRP (universal software
radio peripheral) [40] using GNU radio [16] and TI CC1310 [6] (Figure 2). A
dual-radio USRP connected to Raspberry PI or Laptop is used as the BS. A
CC1310 device or a single-radio USRP can be used as a SNOW node. CC1310 is a
tiny, cheap ($<$$30), and commercially off-the-shelf (COTS) device with a
programmable PHY. We have adopted the open-source implementation of SNOW that
is available at [47].

##  4 System Model

We consider multiple independent and uncoordinated SNOWs. Each SNOW is having
its own BS and associated nodes. The nodes are battery-powered and thus have
energy constraints. We assume the existence of both mobile and stationary
nodes in SNOW. A mobile node can move from one SNOW to any SNOW, as depicted
in Figure 3. Since the BS has a long-range, it can cover a wide area. Hence,
we assume the BSs are stationary. Each node is equipped with a half-duplex
white space radio. The BS and its associated nodes form a star topology where
nodes can directly communicate with the BS.

Figure 3: Inter-SNOW mobility: the figure shows multiple SNOWs where a mobile
node is moving from one SNOW to another.

The BSs are independent, connected to the Internet, and directly connected to
a power source. Each BS uses a wide channel. This channel is split into
narrowband channels/subcarriers of equal width. Each node is assigned a single
subcarrier for transmission and reception to/from the BS. The nodes are kept
simple by offloading the complexities to the BS. The BS determines the
availability of white space at its location by querying a cloud-hosted
database through the Internet. We assume each BS knows the location of its
associated nodes either manually or through existing localization techniques
[28]. However, we are not considering localization in this paper.

##  5 Handling Mobility

In this section, we present our techniques to address mobility challenges in
both intra- and inter-SNOW mobility. We propose to address those through
lightweight cross-layer approaches (MAC-PHY design) feasible at energy-
constrained nodes. First, we present our mobility handling within the same
SNOW (intra-SNOW mobility), and then we will present mobility handling across
SNOWs (inter-SNOW mobility).

###  5.1 Handling Mobility within the Same SNOW

Mobility affects communication reliability even when a node moves within the
same network due to ICI occurred in OFDM subcarriers and also due to
geospatial variation of spectrum within the same network. We address both
scenarios as described below.

####  5.1.1 Handling Mobility Impacts on ICI

ICI is introduced mainly due to the CFO, which stems from the frequency
mismatch between the transmitter and receiver oscillators due to hardware
imperfections and the Doppler shift which is a function of their relative
speed. In SNOW, the subcarriers loose their orthogonality due to such CFO as
shown Figure 4. Hence, to improve an OFDM system’s performance, CFO needs to
be estimated and compensated. Currently, in SNOW, CFO estimation and
compensation is done considering stationary nodes or assuming node speeds are
known [39]. However, SNOW nodes are energy-constrained and low-cost and may
not be equipped to determine speeds. We first give an overview of the adopted
CFO estimation technique and then describe our Doppler shift handling approach
without the need to know the speed of the nodes or under non-uniform speed.

Figure 4: The impact of CFO on subcarriers orthogonality.

SNOW uses training symbols (preamble) for CFO estimation. Due to its
distributed and asynchronous nature, CFO estimation in D-OFDM is done slightly
differently than traditional OFDM. CFO estimation in D-OFDM is done when a
node joins the network. SNOW uses one (or more) subcarrier for a node joining
the network, called join subcarrier, that does not overlap with any other
subcarrier. Each node joins the network by first communicating with the BS on
a join subcarrier. Each way, communication (BS to node and node to BS) follows
a preamble used to estimate CFO on join subcarrier. Specifically, preamble
from a node to BS allows to estimate CFO at the BS, and that from BS to a node
allows to estimate CFO at the node on the join subcarrier. Later, based on the
CFO on a join subcarrier, the CFO on a node’s assigned subcarrier is
determined. CFO estimation technique for both upward and downward
communication is similar. However, CFO compensation approaches in upward and
downward communication are different (refer to [39] for detailed explanation).

First, we explain how CFO is estimated on a join subcarrier $f$. Since it does
not overlap with other subcarriers, it is ICI-free. If $f_{Tx}$ and $f_{Rx}$
are the frequencies at the transmitter and at the receiver, respectively, then
their frequency offset $\Delta f=f_{Tx}-f_{Rx}$. For transmitted signal
$x(t)$, the received signal $y(t)$ that experiences a CFO of $\Delta f$ is
given by

$y(t)=x(t)e^{j2\pi\Delta ft}$ (1)

$\Delta f$ is estimated based on short and long preamble approach using time-
domain samples. A 32-bit preamble is divided into two equal parts, each of 16
bits. First part is for coarse estimation and the second part is for finer
estimation of CFO [49]. Considering $\delta t$ as the short preamble duration,

$y(t-\delta t)=x(t)e^{j2\pi\Delta f(t-\delta t)}.$

Since $y(t)$ and $y(t-\delta t)$ are known at the receiver,

$\displaystyle y(t-\delta t)y^{*}(t)$ $\displaystyle=x(t)e^{j2\pi\Delta
f(t-\delta t)}x^{*}(t)e^{-j2\pi\Delta ft}$
$\displaystyle=|x(t)|^{2}e^{j2\pi\Delta f-\delta t}$

Taking angle of both sides,

$\sphericalangle y(t-\delta t)y^{*}(t)=\sphericalangle|x(t)|^{2}e^{j2\pi\Delta
f-\delta t}=-2\pi\Delta f\delta t.$
$\text{Thus,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}\Delta
f=-\frac{\sphericalangle y(t-\delta t)y^{*}(t)}{2\pi\delta
t}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

A SNOW node calculates the CFO on join subcarrier $f$ using the preambles from
the BS to the node using the above approach. In upward communication, the
time-domain samples are used for CFO estimation on the join subcarrier $f$ at
the BS based on the above approach. Then the ppm (parts per million) on the
receiver’s (BS or SNOW node) crystal is given by
$\text{ppm}=10^{6}\frac{\Delta f}{f}$. Thus, the receiver (BS or a node)
calculates $\Delta f_{i}$ on subcarrier $f_{i}$ as

$\Delta f_{i}=\frac{f_{i}*\text{ppm}}{10^{6}}.$

Thus the BS and a SNOW node that is assigned subcarrier $f_{i}$ calculates CFO
on $f_{i}$ on its respective side. As the nodes asynchronously transmit to the
BS, doing the CFO compensation for each subcarrier at the BS is quite tricky.
Hence, a simple feedback approach for proactive CFO correction in upward
communication is adopted. In this approach, a transmitting node adjusts its
frequency based on $\Delta f_{i}$ when transmitting on subcarrier $f_{i}$ so
that the BS does not have to compensate for $\Delta f_{i}$.

Since mobility causes Doppler shift in frequency contributing further to CFO,
CFO has to be estimated using the above approach while a node moves. If a node
moves at speed $v$, such Doppler Frequency Offset (DFO), denoted by
$\delta_{f}(v)$, is upper-bounded by

$\delta_{f}(v)=\dfrac{v}{c}f_{c}$ (2)

where, $c$ is the speed of light, and $f_{c}$ is the carrier frequency.
Therefore, considering $\Delta f_{i}$ as the CFO when a node is stationary, it
experiences a total CFO of $\Delta f_{i}+\delta_{f_{i}}(v)$ when it moves at
speed $v$. Therefore, to account for this total CFO, the node needs to know
its speed. Besides, when the speed changes, $\delta_{f_{i}}$ has to be
recalculated. But, being energy-constrained and low-cost, SNOW nodes are not
equipped to determine their speeds. Hence, we rely on the observation that
when CFO is estimated for a moving node using the above CFO estimation
technique, its estimation includes both $\Delta f_{i}$ and $\delta_{f_{i}}$,
resulting in a CFO of $\Delta f_{i}+\delta_{f_{i}}$. Thus, the node does not
need to know its speed. If the node’s speed changes, then the total CFO
changes and we need re-estimate. However, the node has no way to determine if
its speed increases or decreases. To handle this challenge, we enable each
node to periodically estimate the CFO. This period can be set as a tunable
system parameter that can be adjusted dynamically. Estimating CFO periodically
will ensure that if the speed changes, the new CFO calculation takes the new
speed into account.

(a) Subcarrier aligned with BS (b) Scenarios showing unalignment

Figure 5: Alignment between the node subcarrier and BS subcarrier.

####  5.1.2 Handling Mobility Impacts on Geospatial Variation of White Spaces

Due to long range, a node’s mobility even within the same network affects the
spectrum availability. For example, a subcarrier that is assigned to a node at
a particular place may not be available if the node moves to another location
within the range of the same BS (i.e., within the same SNOW). Currently, the
BS assigns subcarriers to the nodes without considering their mobility. This
may affect the communications of the mobile nodes. Namely, if a node is highly
mobile and may move anywhere inside the network but is assigned a subcarrier
which is available only in a few locations, its subcarrier assignment is not
much useful. To handle this problem due to geospatial variation of white
spaces, we propose a mobility-aware subcarrier assignment policy as follows.

Note that the BS is already assumed to know the location information of its
coverage area. We also assume that the BS knows the degree or rate of mobility
of each node (i.e, how much mobile the node is). A node can provide a rough
estimate of its mobility when it joins the network. For example, in
agricultural IoT, the system designer knows the number of mobile nodes (e.g.,
tractors and drones) and each node’s mobility rate in a specific geographical
area. The BS orders the nodes based on their mobility, where the stationary
nodes come first and the most mobile node is the last. The BS then orders the
subcarriers based on their availability, from the least widely available
(inside its communication range) subcarrier to the most widely available one.
That is, the subcarrier that is available in the minimum number of locations
comes first and that available in the maximum number of locations (inside the
network) comes at the last of this order. If there are $m$ subcarriers and $n$
nodes, each subcarrier is roughly shared by $\lceil\frac{n}{m}\rceil$ nodes.
Starting from the beginning of the ordered subcarriers, each subcarrier is
then assigned roughly to $\lceil\frac{n}{m}\rceil$ nodes that are not yet
assigned a subcarrier starting from the beginning of the ordered nodes. In
this way, we ensure that the widely available subcarriers are assigned to
highly mobile nodes and the least widely available subcarriers are assigned to
stationary or less mobile nodes.

###  5.2 Handling Mobility across SNOWs

In this section, we present our approach to addressing the mobility across
different SNOWs. Specifically, we handle mobility problem that arises when a
node goes out of the range of a BS. When a node goes out of the range of a BS,
it needs to discover a new BS and get associated with it. Handoff becomes an
issue when a node moves to an uncoordinated SNOW whose operating spectrum is
unknown.

For SNOW, the white space range is very wide, and the SNOW BS may be using a
channel anywhere in that spectrum. A node operates on a narrowband subcarrier.
Two subcarriers at center frequencies $f_{i}$ and $f_{j}$, $f_{i}\not=f_{j}$,
are orthogonal when over time $T^{\prime}$ [7]:

$\int_{0}^{T^{\prime}}\cos(2\pi f_{i}t)\cos(2\pi f_{j}t)dt=0.$ (3)

For example, when the overlap between subcarriers is 50%, the BS bandwidth is
6 MHz, and the subcarrier width is 200 kHz, we can have 59 orthogonal
subcarriers. Thus, to discover a new BS, it is very energy and time consuming
for a low-power node as the node may need to scan thousands of subcarriers.
Our approach has to deal with the following challenges as well. (1) Spectrum
dynamics due to primary user activity is handled using backup subcarriers in
SNOW. However, such an approach does not work under mobility as the backup
channels may be unavailable in a new location. A SNOW node has no access to
the database and thus does not know the white space spectrum availability in
its location. Spectrum sensing is highly energy consuming and is not feasible
for it. (2) It cannot transmit any probing message to explore a BS as it can
interfere with primary users. The node hence needs to depend only on listening
to SNOW’s communication. (3) The nearby BS may be using subcarriers of
different bandwidth, and thus the node subcarrier may be unaligned (as
depicted in Figure 5) and listening to nothing. Aligning with a BS channel is
quite difficult as the BS subcarrier bandwidth is unknown to the moving node.
(4) The node should be able to distinguish between a primary user and a
secondary user (BS). Our steps to address these challenges are as follows.

####  5.2.1 BS Discovery

A direct approach to minimizing BS discovery overhead is that the current BS
can provide a node, before it moves, the channels that the BS would find at 8
locations $(0,\pm r),(\pm r,0),(\pm r,\pm r)$, considering its (estimated)
communication range $r$ and location at $(0,0)$ assuming a Cartesian plane as
shown in Fig. 6. After a node moves out of the current BS range, it can scan
only those channels to find a neighboring BS. However, this approach would
only work if it can inform the BS of its intention to move before it starts to
move. Second, the node needs to know the direction of its movement and inform
the BS. Hence, we also propose another energy-efficient and fast BS discovery
technique that does not depend on these requirements. It utilizes the spectrum
information by combining the received signal features to distinguish between
primary users (TV stations) and a SNOW BS, and considers the trade-off between
discovery latency and energy consumption to allow efficient handoff
management.

Figure 6: Channel availability information in 8 locations.

After a node goes out of range of its BS, it will scan one or more of its
subcarriers. In either case, if it senses signal strength on a subcarrier,
then it has to determine whether it is a BS or a primary user. If its
subcarrier is not aligned with that subcarrier (in case of BS), it may not
decode the received packets. To distinguish between the TV signal and the BS
signal, we first need to detect the presence of primary users. The FCC
regulation for protecting primary incumbents define a protection contour for
TV station as the area where the received signal strength (RSS) is $>-84$dBm
[34]. We follow an approach similar to the one presented in Waldo [41].
Waldo’s results show that low-cost sensors can efficiently detect white spaces
ignored in the databases and existing approaches. Furthermore, depending on
the white space device’s antenna height, further separation (6 km for portable
devices) is required to protect the primary incumbent. To detect white spaces,
FCC recommends a typical antenna height of 10 meters. We consider an antenna
height of 2 meters and compensate for the difference (8 meters) using the
antenna correction factor using Hatas’ urban area propagation model [41]
considering $h_{m}$ as the antenna height in meters as follows.

$a(h_{m})=3.2(\log 11.5h_{m})^{2}-4.97$ (4)

Using Hata’s model, the calculations result in $a(h_{m})=7.5$dB, which will be
added uniformly to the RSS measurements. The addition of the antenna
correction factor directly impacts the noisy measurements by making it closer
to the threshold. Hence, improving the probability of false TV channel
detection. We also follow Waldo’s approach by considering the location safe
for white space operation if the RSS $\geq-84$dBm, and the nearest measurement
is 6 km away [41]. We record the measurements in a large metropolitan city for
five different TV channels (14, 22, 33 are occupied by TV stations, 16 and 25
are white spaces). Additionally, we use the spectrum analyzer measurements and
Google spectrum database as the ground truth to evaluate the SNOW nodes’ TV
channel detection performance. We collect 1500 spectrum measurements using
four low-cost SNOW nodes (TI CC1310 [6]) over a period of 48 hours. Figure 7
shows the results for TV station detection. It is clear from Figure 7(a) that
without considering the antenna correction factor, CC1310 fails to detect TV
transmission in all occupied channels. Operating in a low-frequency spectrum
gives SNOW a tremendous obstacle penetration performance [42]. Additionally,
extensive experiments on TV detection performance is found in [41].

(a) Channel detection before adding antenna correction factor (b) Channel
detection after adding antenna correction factor

Figure 7: Performance of TV detection

We utilize a number of features of the received signals to distinguish between
primary users (TV stations) and a SNOW BS. In addition to the common
observation that RSS of the TV transmission is high and the signal amplitude
is constant, primary user communication is observed to be continuous over a
long duration (see Figure 8(a)). In contrast, SNOW BS signal amplitude is
fluctuating during transmission and the BS may not have continuous
communication for long periods as shown in Figure 8(b). In addition, if
multiple consecutive channels have similar RSS, it is likely to be a BS
because a BS typically uses more than one TV channel. For primary users, two
consecutive channels should belong to two different primary users, and their
signal strengths on two consecutive channels should be a lot different. To
enable faster discovery, we also consider using a wider band for sensing,
which will enhance the BS detection probability but will consume more energy.
Since using a narrow subcarrier for searching can take a longer time, thus
consuming much energy, such tradeoff is left as a design choice.

(a) TV transmission

(b) SNOW transmission

Figure 8: TV and SNOW signal transmission recorded using a spectrum analyzer.

####  5.2.2 Subcarrier Alignment

Different SNOWs can have different subcarrier bandwidth, e.g., camera or audio
may use a wider subcarrier. Thus upon discovery of a new BS, as shown in
Figure 5, a node’s subcarrier may not be aligned with a BS subcarrier.
Alignment is needed to start communication. Existing channel rendezvous
techniques are not applicable as they consider the channels of equal
bandwidth. Thus, this problem is specific to SNOW. By Equation (3), an overlap
can start from many points of a nearby subcarrier. Such an overlap makes the
problem highly challenging. To solve the problem, we exploit several
characteristics of SNOW design. Even though SNOW can use any subcarrier
bandwidth, we consider that subcarrier bandwidth does not vary arbitrarily,
and we assume each BS uses a subcarrier bandwidth from the values 100kHz,
200kHz, 400kH, or 600kHz. Upon discovering the presence of a BS, this
assumption helps us simplify the synchronization with its subcarrier. This
will be done using a wider bandwidth at the node and combining time and
frequency domain energy-sensing.

The time-domain sensing is the typical carrier sensing that calculates the
energy level using a moving average of the digital signals, i.e., the sequence
of discretized, complex samples from the analog-to-digital converter, within a
short period. A channel is considered busy if the output exceeds the
predefined threshold. The moving average’s window size is set to half of the
length of the preamble to ensure prompt sensing of a packet. Although time-
domain sensing alone can sense a busy channel, it does not distinguish between
different subcarriers. A node needs to analyze the frequency domain of the
signals further. Specifically, it calculates the power spectrum density (PSD)
of the recent $M$ samples using Fast Fourier Transformation (FFT). The node
analyzes the power distribution and compares it with all possible channel-
overlapping patterns based on the PSD. Intuitively, if the power is uniformly
distributed over the entire spectrum, then the signals on the air come from a
fully-overlapped subcarrier; otherwise, only a fraction of the channel is
occupied. The exact fraction of channel in use is hard to calculate because
different subcarriers may exhibit different power levels due to frequency-
selective fading, and the imperfect hardware filter (used to confine the
radio’s bandwidth) spreads over the boundary of the PSD curve. But considering
a limited number of bandwidths, the node can explore possible overlapping
patterns and select the one with maximum matching with the PSD. The number of
such patterns will also be limited as it is done after determining the
presence of a BS.

In SNOW, a node is less powerful and energy-constrained. The complexity of
time-domain sensing is the same as the RSSI calculation in typical
communications systems, which is linear with respect to the number of incoming
samples. Since frequency sensing is performed only after a sequence of signals
pass the time domain sensing, it takes constant time irrespective of the
number of samples. The constant depends on the number of packets that cause
the time-domain sensing to return busy. Note that such an approach is needed
only when a node moves to an uncoordinated SNOW. Once the node is aligned with
any subcarrier of the new BS, it can use CSMA/CA approach to transmit to the
BS and ultimately join the network. Figure 9 shows the subcarrier alignment
latency for different subcarrier bandwidths.

Figure 9: Subcarrier alignment latency

##  6 Experiments

We have first implemented our mobility approaches using TI CC1310 devices as
SNOW nodes. TI CC1310 is a tiny, low-cost, and low-power COTS device with a
programmable PHY which was recently adopted as SNOW node [39]. To perform
experiments at much longer communication ranges, we have also implemented our
mobility approaches using USRP devices as SNOW nodes based on its current
open-source implementation in GNU Radio [47]. GNU Radio is an open-source
development toolkit that provides signal processing blocks to develop
software-defined radio [16]. USRP is a hardware platform designed for RF
application [40]. We have used two USRP 210 devices, each having a dual radio,
as two BSs in each experiment for inter-SNOW mobility experiments. In the
first set of experiments, we have used 10 TI CC1310 devices as SNOW nodes. In
the other set of experiments, we have used 7 USRP 200 devices, each with a
single radio, as SNOW nodes. The USRP devices operate in the band 70MHz –
6GHz. Packets generation, decoder, and other implementation are adopted from
SNOW open source implementation [47].

Note that our experiments are performed mainly considering inter-SNOW mobility
to show that our approach can enable such mobility. We cannot compare the
results against the scenario when our approaches are not adopted because
inter-SNOW mobility cannot be enabled without our approaches. However, we
compare the performance against the stationary scenario to observe the
performance degradation under mobility. In our experiments we shall
demonstrate that such degradations are not high and our approaches show
robustness in terms of reliability, latency, and energy consumption under
various mobility scenarios.

###  6.1 Default parameters

Parameters of interests are calibrated in different experiments based on
requirements and the rest are left as defaults. The default experimental
parameter settings are as follows.

  * •

Frequency band: varying (470 MHz - 599 MHz)

  * •

Modulation: ASK/OOK

  * •

Packet size: 40 bytes

  * •

BS bandwidth: 6 MHz

  * •

Node bandwidth: TI CC1310: 200 kHz, USRP: 400 kHz

  * •

TX power: TI CC1310: 15 dBm, USRP: 0 dBm

  * •

Receiver sensitivity: -110 dBm

  * •

Distance: Indoor: 10 - 50 m, Outdoor: 900 m

###  6.2 Experiments with TI CC1310: Indoor and Outdoor Deployment

####  6.2.1 Indoor Deployment

The experiments with CC1310 were carried out in a hallway on the third floor
inside the computer science building at Wayne State University. We fixed the
position of the BSs while a person is continuously moving at average walking
speed from one end of the hallway to the other for 30 minutes. We kept the
antenna height at 2 meters above the ground for all experiments. In all the
experiments, the CFO and CSI are estimated and compensated based on SNOW
implementation in [39]. We used the default setting for all the experiments.

Reliability under Mobility. We kept the distance between the node and BSs at
approximately 10 meters to observe our proposed mobility approach’s
reliability. One node is stationary at this distance, and another node is
continuously moving from one BS towards the other. The stationary node
transmits 5000 packets to the BS while the mobile node transmits 2500 packets
to the first BS and 2500 to the second BS after the joining process. The
results in Figure 10(a) demonstrate that with minor human mobility, the Packet
Error Rate (PER) slightly increases under our approach. For the stationary
node, the PER is around 0.02%, while the mobile node is 0.72%. Also, we
observe reliability with varying packet sizes. Figure 10(b) demonstrates the
impact of packet size on our mobility approach. With 20-byte packet, the
stationary node PER is 0% (no packet loss). For the same packet size, the
mobile node PER is around 4.5%. Furthermore, for the 40-byte packet, the PER
is 0.1% and 5.2% for the stationary and mobile node. With a 100-byte packet,
the mobile node achieves 5.4% PER, while the PER for the stationary node is
0.39%. This result shows that packet size has an impact on reliability. Larger
packets require more air time to receive, resulting in more interference
leading to increased PER. The difference in the reliability performance
between the two results is due to the channel condition changing. For the
mobile node, moving from one BS to another might increase the PER due to the
channel condition at the new location, which might increase the PER. However,
the results prove that our approach offers reliable communication under
mobility.

(a) Reliability under mobility (b) Reliability with varying packet size

Figure 10: Reliability under mobility and varying packet size.

Maximum Achievable Throughput. The maximum achievable throughput is the total
maximum number of bits the BS can receive per second. In this experiment, we
calculate the maximum achievable throughput using our approach compared to the
stationary nodes. In both scenarios, each node transmits 1000 40-byte packets.
Figure 11 shows that in a stationary scenario, the maximum achievable
throughput is 240 kbps compared to 174 kbps during mobility when ten nodes
transmit simultaneously. This result is not surprising since mobility
increases the packet loss rate, which affects the throughput.

Figure 11: Throughput with varying # of nodes.

Energy Consumption and Latency. To estimate the energy consumption and latency
of CC1310 during mobility, we measure the average energy consumed at the nodes
and the time it takes to transmit 1000 packets per node successfully. We
placed ten nodes, each 50 m away from BS1. We performed two sets of
experiments (stationary nodes and mobile nodes). In the mobility experiment,
the nodes are placed 10 m away from BS1 and 50 m away from BS2. And the nodes
move from BS1 to BS2. Also, we measure the overhead of the BS discovery and
subcarrier alignment offline and add the results accordingly. To calculate the
energy consumption, we use the energy model of CC1310 (Voltage is 3.8v, RX
5.4mA, and TX 13.4mA). We measure the time required to collect 10.000 packets
at BS1 for the stationary nodes and 2500 packets and 7500 packets at BS1 and
BS2, respectively, in the mobile scenario. Figure 12 shows that in a
stationary scenario, the average energy consumed by the node is 81.4mJ
compared to 87.1mJ in during mobility. We observe similar behavior for the
latency. As shown in Figure 13, the average latency for stationary nodes is
1.6s compared to 1.712s in mobile nodes. This result indicates that the number
of nodes has an insignificant impact on the energy and latency regardless of
mobility. This is due to the capability of SNOW BS, which allows multiple
nodes to transmit in parallel.

Figure 12: CC1310 Energy consumption Figure 13: CC1310 Latency

####  6.2.2 Outdoor Deployment

In this experiment, we evaluate our mobility approach’s performance in the
city of Detroit, Michigan (see Figure 17), in terms of maximum achievable
throughput, energy consumption, and latency using CC1310 devices in outdoor
deployments. We fix the location of the BSs and place the node inside a moving
vehicle. The distance between the node and the BSs is approximately 900
meters. The vehicle speed varies between 5 mph and 40 mph. The data is
collected at the BSs for further analysis.

Maximum Achievable Throughput. In this experiment, we compare the maximum
achievable throughput at different speeds (5 mph, 20 mph, 40 mph). Each node
transmits 1000 40-byte packets, and we calculate the combined throughput at
the BSs. Figure 14 shows that the maximum achievable throughput is
approximately 12.5 kbps at 5 mph speed compared to 11.89 kbps and 11.83 kbps
At 20 mph and 40 mph, respectively. This shows that the speed (up to 40 mph)
has an insignificant impact on the nodes’ maximum achievable throughput.

Figure 14: Throughput vs. node speed

Energy Consumption and Latency. To estimate the energy consumption and latency
in outdoor deployment, we place the BSs at varying distances from the nodes
(up to 900 meters). We set the nodes inside a moving vehicle. We measure the
average energy consumed at each node and the time it takes to transmit 1000
packets per node successfully. Figure 15 shows that the average energy
consumed by the node moving 5 mph is to 87.1mJ and 87.3 mJ at 100 m and 900 m,
respectively. At 20 mph, the energy consumption is 87.1 mJ and 87.4 mJ at 100
m and 900 m, respectively. The average energy consumption is 87.8 mJ and 87.9
mJ at 100 m and 900 m for 40 mph. These results are similar to the indoor
scenario where the node is moving at walking speed. In Figure 16, the average
latency for all nodes, regardless of the distance, is 1.735s. This result
shows that the energy and latency for all nodes are similar except for the
energy consumption at a 40 mph speed and a distance of 900 m where the results
slightly vary due to higher PER.

Figure 15: CC1310 Outdoor energy consumption Figure 16: CC1310 Outdoor latency

###  6.3 Experiments with USRP: Deployment in a Metropolitan City

Figure 17 shows the distances in Detroit city in Michigan, where the mobility
data were collected from nodes placed inside a moving car. The BSs are kept
stationary. The vehicle is continuously moving at varying speeds (up to 40
mph) from one BS to the other in the mobile scenario. The antenna height was
kept at 2 meters above the ground in all the experiments. We used the default
setting in all the experiments.

Figure 17: USRP experimental setup

####  6.3.1 Reliability over Distance

To observe the effect of distance on the reliability of SNOW in mobile
scenarios, we collect the data at 300m, 500m, 700m, and 900m from the BS,
respectively. Each node transmits 5000 packets. To measure the reliability, we
chose Correctly Decoding Rate (CDR), which is the ratio of the number of
correctly decoded packets at the BS to the total number of transmitted packets
[43]. Figure 18 shows the reliability over various distances from the BS when
the node is moving from one BS to the other. At 300 meters, the BSs can decode
on average 96.6% of the packets from the mobile node compared to 100% for the
stationary node. Furthermore, at 500 meters away, the mobile node’s
reliability reduces to 96%, while the stationary node achieves 99.99%
reliability. At 900 meters, the reliability is 80% for the mobile node
compared to 99.95% at the stationary node. These results show that the
distance between the mobile node and BS has a significant impact on decoding
reliability. However, Even for the stationary node, its’ performance is
slightly impacted by the distance from the BS.

Figure 18: Reliability over distances

####  6.3.2 Performance of SNOW with CFO

In this experiment, we observe the performance of SNOW to demonstrate the
effect of CFO estimation and compensation in mobile environments. We compare
the CDR of a mobile SNOW node in two cases, with CFO estimation and
compensation and without CFO compensation. Also, we compare the performance of
each case to the performance of a stationary SNOW node. All the nodes were
500m away from the BS. The mobile nodes were placed in a car moving at varying
speeds. Each node transmits 5000 packets asynchronously to the BSs. For mobile
nodes, each node transmits 2500 packets to BS1 and 2500 to BS2. Figure 19
demonstrates the effect of CFO under mobility. For stationary nodes, the
average CDR is around 99.97% for all the transmitted packets. Without
compensation for CFO, the average CDR is around 80% for all the nodes.
However, we compensate for CFO; the average CDR increases to 96%, which is
significant. This result demonstrates that in mobile environments, CFO could
severely impact the transmission reliability. Thus, CFO estimation and
compensation could significantly increase the reliability of the transmission
in inter-SNOW communication.

Figure 19: Performance under mobility with CFO

####  6.3.3 Maximum Achievable Throughput

In this experiment, we compare the maximum achievable throughput in mobile
inter-SNOW with the stationary SNOW. For both scenarios, each node transmits
100 40-byte packets. We calculate the combined throughput at the BSs. Figure
11 shows that when 8 nodes are transmitting, the maximum achievable throughput
is 298 kbps and 393 kbps for mobile and stationary SNOWs, respectively. During
mobility when 10 nodes transmit simultaneously. Due to the increased packet
loss rate during mobility, stationary SNOW achieves better throughput.

Figure 20: Throughput vs # of node

####  6.3.4 Energy Consumption and Latency

In this experiment, we demonstrate the efficiency of our mobility approach for
USRP in terms of energy consumption and latency. Specifically, we compare the
efficiency of mobile SNOW with stationary SNOW. We observed that the
performance of SNOW under mobility is affected by the distance from the BS.
Hence, for a fair comparison with stationary SNOW, we place 7 mobile node 900m
away from the BS2 while continuously moving at approximately 20mph towards
BS1. Furthermore, since USRP devices allow for bidirectional communication,
each node transmits 100 packets (50 to BS1 and 50 to BS2 during mobility)
during the upward duration (1s) and waits until the end of the upward duration
to receive an acknowledgment (ACK) from the BSs. We then calculate the average
energy consumption per node and the time needed to collect all the packets at
the BS.

Figure 21 shows that the average energy consumed at the mobile nodes is around
47.4mJ compared to 47.32% in stationary nodes when 7 nodes transmit. This
shows that mobility has a minimal impact on the energy efficiency of the node.
Similar to the average energy consumption, Figure 22 shows that the latency of
collecting all packets in mobile SNOW is comparable to the stationary SNOW.
These results demonstrate that the efficiency of SNOW is not affected by
mobility.

Figure 21: Energy consumption Figure 22: Latency

##  7 Conclusions

In this paper, we have addressed mobility in SNOW (Sensor Network Over White
spaces), an LPWAN that is designed based on D-OFDM and that operates in the TV
white spaces. SNOW supports massive concurrent communication between a base
station (BS) and numerous nodes. We have proposed a dynamic CFO estimation and
compensation technique to handle mobility impacts on ICI. We have also
proposed to circumvent the mobility impacts on geospatial variation of white
space through a mobility-aware spectrum assignment to nodes. To enable
mobility of the nodes across different SNOWs, we have proposed an efficient
handoff management through a fast and energy-efficient BS discovery and quick
association with the BS by combining time and frequency domain energy-sensing.
Experiments through SNOW deployments in a large metropolitan city and indoors
have shown that our proposed approaches enable mobility across multiple
different SNOWs and provide robustness in terms of reliability, latency, and
energy consumption under mobility.

##  8 Acknowledgments

This work was supported by NSF through grants CAREER-1846126, CNS-2006467, and
by Wayne State University through the Rumble Fellowship.

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# On the existence of extreme coherent
distributions with no atoms

Stanisław Cichomski  Department of Mathematics, Informatics and Mechanics
University of Warsaw
Banacha 2, 02-097 Warsaw
Poland  and  Adam Osękowski  Department of Mathematics, Informatics and
Mechanics
University of Warsaw
Banacha 2, 02-097 Warsaw
Poland

###### Abstract.

The paper is devoted to the study of extremal points of $\mathcal{C}$, the
family of all two-variate coherent distributions on $[0,1]^{2}$. It is well-
known that the set $\mathcal{C}$ is convex and weak∗ compact, and all extreme
points of $\mathcal{C}$ must be supported on sets of Lebesgue measure zero.
Conversely, examples of extreme coherent measures, with a finite or countable
infinite number of atoms, have been successfully constructed in the
literature. The main purpose of this article is to bridge the natural gap
between those two results: we provide an example of extreme coherent
distribution with an uncountable support and with no atoms. Our argument is
based on classical tools and ideas from the dynamical systems theory. This
unexpected connection can be regarded as an independent contribution of the
paper.

##  1\. Introduction

The concept of coherent random variables and distributions appeared for the
first time in the mid-nineties, in the paper by Dawid et al. [17]. Within a
Bayesian inference model, they investigated practical procedures for combining
distinct conditional probabilities of a fixed uncertain event into a single
and reliable forecast. Honoring their original definition, we say that a
multivariate random vector $(X_{1},X_{2},\dots,X_{n})$ defined on some
arbitrary probability space $(\Omega,\mathcal{F},\mathbb{P})$ is coherent, if
there is an event $E\in\mathcal{F}$ and sequence of sub-$\sigma$-fields
$\mathcal{G}_{1},\mathcal{G}_{2},\dots,\mathcal{G}_{n}\subset\mathcal{F}$ such
that

(1.1) $X_{i}=\mathbb{P}(E|\mathcal{G}_{i})\ \ \ \ \text{for all}\ \ \ \
i=1,2,\dots n,\ \ \ \ \text{almost surely}.$

The motivating interpretation is as follows: suppose that a group of $n$
experts provides their personal estimates on the likelihood of an uncertain
event $E$ of our interest. Moreover, assume that the entire knowledge of
$j$-th specialist ($j=1,2,\dots,n$) is strictly restricted to some
personalized source of information (sub-$\sigma$-algebra) $\mathcal{G}_{j}$
and his predictions are solely based on this limited knowledge. Then, in such
an idealized model, the obtained sequence $(X_{1},X_{2},\dots,X_{n})$ of
experts’ opinions on the odds of event $E$, clearly satisfies (1.1). In such a
case, the joint distribution of vector $(X_{i})_{i=1}^{n}$ on the unit cube
$[0,1]^{n}$ is also called coherent. In what follows, we write
$m\in\mathcal{C}_{n}$ to indicate that a probability measure $m$ is the
distribution of some $n$-variate coherent vector $(X_{i})_{i=1}^{n}$.

After two decades of hiatus, the mathematical study of coherence has been
reinstated by the influential paper of Burdzy and Pitman, see [9]. For the
latest advancements in the study of multivariate coherent distributions, see
e.g. [12, 13, 19]. Modernly, the notion of coherent distributions is well
suited to the analysis of information diverisity in social networks and
stochastic modelling of other data-driven technologies. Quite unexpectedly, it
fits efficiently into the rapidly growing framework of statistical learning,
game theory and microeconomics. More specifically, we refer the interested
reader to the works on optimal combining [18, 27, 29], posterior feasibility
[2, 4, 14], Bayesian rationality [5, 22, 26], maximal discrepancy [8, 12, 16]
and Bayesian persuasion [1, 3, 25].

Just as importantly as its applications in statistics and microeconomics, the
notion of coherent distributions is a valuable mathematical concept in its own
right. Its distinctive feature is that it appears at the intersection of two
profound classical fields: probability theory and discrete mathematics. On the
purely probabilistic side, coherent distributions are closely related to
permutons and copulas [15, 24, 32], study of probabilistic measures with given
marginals [3, 21, 30], or theory of martingales indexed by partially ordered
sets [9, 14, 19]. From the combinatorial perspective, coherent distributions
are strongly connected with graph theory [11, 16, 31], mathematical tomography
[6, 21, 23] and combinatorial matrix theory [7, 11, 28]. Our work here reveals
an unexpected connection with another field of mathematics: below, we will
exhibit the link between the geometric analysis of coherent distributions with
the theory of dynamical systems.

Although most of the existing literature deals with some various optimization
problems, ongoing research is strictly constricted by the fact that geometric
structure of coherent distributions is very complex and not entirely explored.
Let us briefly recall what is known. From now on, we will drop the subscript
’$2$’ and write $\mathcal{C}$ instead of $\mathcal{C}_{2}$ for the
foundational class of all two-dimensional coherent distributions on
$[0,1]^{2}$. As demonstrated in [9], the family $\mathcal{C}$ is a convex and
weak∗ compact subset of all probability distributions on $[0,1]^{2}$. Studying
$\mathrm{ext}(\mathcal{C})$, the set of extreme points of $\mathcal{C}$, is
consequently the primary challenge in this field. As shown independently in
[4, 32], there exist extreme coherent measures with arbitrary large or even
countable infinite number of atoms. For a useful comparison, note that
martingales $(M_{1},M_{2})$, whose distributions are extremal, are always
supported on one or two points, see [20]. Next, as verified in [4], elements
of $\mathrm{ext}(\mathcal{C})$ are always supported on sets of Lebesgue
measure zero. Lastly, in [15], a complete characterisation of
$\mathrm{ext}(\mathcal{C})$ was developed, but this did not result in any
easily verifiable criterion.

Figure 1. Red color stands for $\Delta_{r}$, the support of measure $\mu_{r}$.
Blue color represents $\Gamma_{r}$, the support of measure $\nu_{r}$. The
union $\Delta_{r}\cup\Gamma_{r}$ is the support of an extreme coherent
distribution $m_{r}$.

The main aim of this article is to tackle the natural and intriguing problem
formulated in [4, 15, 32]: does there exist any non-atomic extreme points of
$\mathcal{C}$? We will answer this question in the affirmative. Beforehand, we
need to develop some auxiliary notation. For $r\in(0,1)$, introduce
$t_{r}:[0,r]\rightarrow[0,r]$ by

$t_{r}(x)=2\min\\{x,r-x\\}$

and distinguish the two special sets

$\Delta_{r}=\left\\{(x,x):x\in[0,r]\right\\}\ \ \ \ \text{and}\ \ \ \
\Gamma_{r}=\left\\{(x,t_{r}(x)):x\in[0,r]\right\\}.$

Next, consider nonnegative Borel measures $\mu_{r}$ and $\nu_{r}$, supported
on $\Delta_{r}$ and $\Gamma_{r}$, respectively, which are defined by

(1.2) $\mu_{r}\left(\left\\{(s,s):0\leq s\leq x\right\\}\right)\ \ =\ \
c_{r}^{-1}\int_{0}^{x}\frac{s}{1-s}\ \mathrm{d}s,$ (1.3)
$\nu_{r}\left(\left\\{\left(s,t_{r}(s)\right):0\leq s\leq x\right\\}\right)\ \
=\ \ c_{r}^{-1}\int_{0}^{x}1\ \mathrm{d}s,$

for $x\in[0,r]$, where

(1.4) $c_{r}\ =\ \int_{0}^{r}\left(\frac{s}{1-s}+1\right)\mathrm{d}s\ =\
-\log(1-r)\ <\infty.$

See Figure 1. Finally, let $m_{r}=\mu_{r}+\nu_{r}$. Our main result is the
following.

######  Theorem 1.1.

For every $r\in(0,1)$, the measure $m_{r}$ is an extreme point of
$\mathcal{C}$.

For notational simplicity, fix any $r\in(0,1)$. Although the choice of $r$ is
arbitrary, we shall not highlight it any further. The main tools and short
outline of the proof are provided in the next section.

##  2\. Main tools and outline of the proof

A few more results and definitions will be needed in order to continue the
discussion. For the sake of clarity, we split the material into a few separate
parts.

###  2.1. A structure of coherent distributions

For a Borel measure $\pi$ on $[0,1]^{2}$, we will write $\pi^{x}$ and
$\pi^{y}$ for the corresponding marginal measures of $\pi$ on $[0,1]$. In
other words, we have

$\pi^{x}(A)=\mu(A\times[0,1])$ and $\pi^{y}(B)=\mu([0,1]\times B)$

for all Borel subsets $A,B\in\mathcal{B}([0,1])$. We begin by recalling the
definition of the family $\mathcal{R}$, as introduced in [15]. This family is
closely related to $\mathcal{C}$ and crucial for our analysis.

######  Definition 2.1.

Let $\mathcal{R}$ be the collection of all ordered pairs $(\mu,\nu)$ of
nonnegative Borel measures on $[0,1]^{2}$, satisfying

$\int_{A}(1-x)\ \mathrm{d}\mu^{x}\ \ =\ \ \int_{A}x\ \mathrm{d}\nu^{x},$

and

$\int_{B}(1-y)\ \mathrm{d}\mu^{y}\ \ =\ \ \int_{B}y\ \mathrm{d}\nu^{y},$

for any Borel subsets $A,B\in\mathcal{B}([0,1])$.

Next, we have the following fact – see [15] or [4] (with a slightly different
formulation).

######  Proposition 2.1.

Let $m$ be a probability measure on $[0,1]^{2}$. Then $m$ is coherent if and
only if there exists $(\mu,\nu)\in\mathcal{R}$ such that $m=\mu+\nu$.

Hereafter, for $m\in\mathcal{C}$ and any pair $(\mu,\nu)\in\mathcal{R}$ with
$m=\mu+\nu$, we will shortly say that $(\mu,\nu)$ is the representation of a
coherent distribution $m$. We have the following fact.

######  Proposition 2.2.

The pair $(\mu_{r},\nu_{r})$ is a representation of a coherent distribution
$m_{r}$.

###### Proof.

We need to check that $(\mu_{r},\nu_{r})\in\mathcal{R}$. First, let us note
that $\mu_{r}^{x}=\mu_{r}^{y}$ since the measure $\mu_{r}$ is concentrated on
the diagonal $\Delta_{r}$. Secondly, we can easily check that
$\nu_{r}^{x}=\nu_{r}^{y}$ and this common measure is proportional to
$\mathcal{U}(r)$, the uniform distribution on $[0,r]$:

(2.1) $(c_{r}/r)\cdot\nu_{r}^{x}\ =\ \mathcal{U}(r),$

where $c_{r}$ is defined as in (1.4). Hence, in the light of Definition 2.1,
we will be done if we verify that

$\int_{A}(1-x)\ \mathrm{d}\mu_{r}^{x}\ =\ \int_{A}x\ \mathrm{d}\nu_{r}^{x},$

for any Borel subset $A\in\mathcal{B}([0,r])$. However, using (1.2) and (1.3),
we get

$\int_{A}(1-x)\ \mathrm{d}\mu_{r}^{x}\ =\
\frac{1}{c_{r}}\int_{A}(1-x)\frac{x}{1-x}\ \mathrm{d}x,$

and

$\int_{A}x\ \mathrm{d}\nu_{r}^{x}\ =\ \frac{1}{c_{r}}\int_{A}x\ \mathrm{d}x,$

which completes the proof. ∎

It is important to distinguish some special types of representations with two
additional significant features. For two Borel measures $\pi_{1}$, $\pi_{2}$
supported on the unit square $[0,1]^{2}$, we write $\pi_{1}\leq\pi_{2}$ if the
estimate $\pi_{1}(A)\leq\pi_{2}(A)$ holds for all
$A\in\mathcal{B}([0,1]^{2})$.

######  Definition 2.2.

Let $m\in\mathcal{C}$. We say that the representation $(\mu,\nu)$ of $m$ is

$\cdot$ _unique_ , if for every $(\tilde{\mu},\tilde{\nu})\in\mathcal{R}$ with
$m=\tilde{\mu}+\tilde{\nu}$, we have $\tilde{\mu}=\mu$ and $\tilde{\nu}=\nu$;

$\cdot$ _minimal_ , if for all $(\tilde{\mu},\tilde{\nu})\in\mathcal{R}$ with
$\tilde{\mu}\leq\mu$ and $\tilde{\nu}\leq\nu$, there exists $\alpha\in[0,1]$
such that $(\tilde{\mu},\tilde{\nu})=\alpha\cdot(\mu,\nu)$.

As carefully explained in [15], these conditions allow to state a complete
description of the class $\mathrm{ext}(\mathcal{C})$.

######  Theorem 2.1.

Let $m$ be a coherent distribution on $[0,1]^{2}$. Then $m$ is extremal if and
only if the representation of $m$ is unique and minimal.

By the above statement, our main result, Theorem 1.1, will be established once
we have shown that representation $(\mu_{r},\nu_{r})$ of $m_{r}$ is indeed
unique and minimal. These two properties will be proved in Section 3 and
Section 4 below.

###  2.2. On the idea behind the special measure $m_{r}$

For $\eta\in[0,2]$, let $T_{\eta}:[0,1]\rightarrow[0,1]$ denote the celebrated
’tent map’ function, defined by $T_{\eta}(x)=\eta\min\\{x,1-x\\}.$ For our
purposes, only the case $\eta=2$ will be needed; the corresponding function
$T_{2}$ is sometimes referred to as a ’full tent map’. After picking a
starting point $x_{0}\in[0,1]$, map $T_{2}$ defines a discrete-time dynamical
system via the recurrence relation

$x_{n+1}\ =\ T_{2}(x_{n})\ =\ \begin{cases}2x_{n}&\mbox{if }\
x_{n}\leq\frac{1}{2},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \displaystyle
2(1-x_{n})&\mbox{if }\ x_{n}>\frac{1}{2},\end{cases}$

for $n=0,1,2,\dots$ Now, if $x_{0}$ is irrational, then the resulting sequence
$(x_{n})_{n=0}^{\infty}$ becomes injective. To describe the dynamics, one
often makes use of the so-called cobweb plot (or Verhulst diagram): put points
successively and alternately on the main diagonal and on the graph of function
$T_{2}$, according to the following algorithm:

  1. 0.

place a starting point $(x_{0},x_{0})$ on the diagonal,

then, for $n=1,2,\dots$, repeat

  1. 1.

add a point $\left(x_{n-1},T_{2}(x_{n-1})\right)$ on the graph of function
$T_{2}$,

  2. 2.

place a point $(x_{n},x_{n})$ on the diagonal and increase $n$ by $1$.

Let us consider a simple extension of this procedure: apart from drawing
points on the diagram, we will also assign specific weights to each new
location. Namely, for a fixed huge integer $N$, place (red) mass
${x_{n}}/({1-x_{n}})$ at every point $(x_{n},x_{n})$ on the diagonal,
$n=1,2,\dots,N-1$, and the (blue) mass $1$ for each point on the graph of
function $T_{2}$. Finally, assign weight zero (and a purple color) to the
initial and terminal points $(x_{0},x_{0})$, $(x_{N},x_{N})$. See Figure 2.

Figure 2. An example of a cobweb diagram showing the first iterations of a
full tent map system. Purple points are endpoints of an ’axial path’ obtained
in this process. Red points represent masses placed on the diagonal, while
blue points indicate masses added on the graph of function $T_{2}$.

We are ready to discuss (informally) the intuition which has led us to the
measure $m_{r}$. Take the initial subsequence
$(x_{0},x_{0}),(x_{0},x_{1}),(x_{1},x_{1}),\dots,(x_{N-1},x_{N})$, for some
big number $N$. Let $\widetilde{m}^{N}$ be a probability distribution
supported on those points, determined by their normalized weights. Then,
choose a random point $P$ on the square (distributed according to
$\widetilde{m}^{N}$) and let $(X,Y)$ be its coordinates. Let $E$ denote the
event that $P$ belongs to the diagonal. Then for any $1\leq i,j\leq N-1$ we
may write

(2.2) $\mathbb{P}(E|X=x_{i})\ =\
\frac{\frac{x_{i}}{1-x_{i}}}{\frac{x_{i}}{1-x_{i}}+1}\ =\ x_{i}$

and

(2.3) $\mathbb{P}(E|Y=x_{j})\ =\
\frac{\frac{x_{j}}{1-x_{j}}}{\frac{x_{j}}{1-x_{j}}+1}\ =\ x_{j},$

where the middle fraction is the ratio of the red mass to the total mass in
the line. Intuitively, equations (2.2) and (2.3) show that $\widetilde{m}^{N}$
is ’almost coherent’: the representation of this measure is the pair of its
red and blue component. Now it is natural to try to carry out a limiting
argument: as $N$ goes to infinity, red points uniformly cover the entire
diagonal and blue points fill up the graph of $T_{2}$. It seems plausible that
$\widetilde{m}^{N}$ converges to a certain limiting measure
$\widetilde{m}\in\mathrm{ext}(\mathcal{C})$ with no atoms. Unfortunately, this
is not the case: as $\frac{x}{1-x}\rightarrow\infty$ for $x\rightarrow 1$, the
red mass explodes uncontrollably near the corner $(1,1)$.

Nevertheless, there is an easy way to overcome this obstacle. Rather than
working with the transform $T_{2}$, we will simply employ its rescaled version
$t_{r}:[0,r]\rightarrow[0,r]$,

$t_{r}(x)=2\min\\{x,r-x\\}.$

Observe that images of transform $T_{2}$ and its shrunken version $t_{r}$ are
homothetic. Thus, if we just repeat the former reasoning with $t_{r}$ acting
on $[0,r]$, instead of $T_{2}$ acting on $[0,1]$, then any point mass added to
the diagram will not exceed $\frac{r}{1-r}<\infty$. This leads us to the
consideration of a new sequence $\widetilde{m}^{N}_{r}$, defined analogously
to $\widetilde{m}^{N}$, of ’almost coherent’ and ’resembling extreme points’
measures on $[0,r]^{2}$. Letting $N\to\infty$, looking at the red and blue
components of $\widetilde{m}^{N}_{r}$ separately, we see that red mass
approaches to $\mu_{r}$ and blue mass tends to $\nu_{r}$. This motivates our
claim: $m_{r}=\mu_{r}+\nu_{r}$ is an extreme coherent distribution.

###  2.3. Some basic facts from the theory of dynamical systems

To carry out a formally strict justification, we will need the following
classical definition of a measure-preserving and ergodic transformation. By
$B\triangle C$ we will denote the symmetric difference $(B\setminus
C)\cup(C\setminus B)$ of sets $B$ and $C$.

######  Definition 2.3.

Fix a constant $s>0$ and let $\pi$ be a Borel probability measure on $[0,s]$.
Let $T:[0,s]\rightarrow[0,s]$ be a Borel measurable function. We say that the
transformation $T$ is

$\cdot$ _measure-preserving_ (or equivalently, that $\pi$ is _$T$
-invariant_), if

$\pi(T^{-1}(A))\ =\ \pi(A),$

for all Borel subsets $A\in\mathcal{B}([0,s])$;

$\cdot$ _ergodic_ (or equivalently, that $\pi$ is an _ergodic measure_), if
$T$ is measure-preserving and any set $A\in\mathcal{B}([0,s])$ with
$\pi\left(T^{-1}(A)\triangle A\right)=0$ must satisfy $\pi(A)\in\\{0,1\\}$.

Therefore, if a measure-preserving transformation $T$ is not ergodic, then we
can find some $A\in\mathcal{B}([0,s])$ with $T^{-1}(A)=A$ and
$\pi(A)\in(0,1)$. In principle, this means that restrictions
$T|_{A}:A\rightarrow A$ and $T|_{A^{c}}:A^{c}\rightarrow A^{c}$ define a non-
trivial decomposition of the map $T$ into two simpler subsystems. Noteworthy,
there is a subtle similarity between ergodicity of transformations and
minimality of representations from Definition 2.2. Eventually, we need to make
more precise the statement: ’full tent map $T_{2}$ covers $[0,1]$ uniformly’.

######  Proposition 2.3.

$\mathcal{U}(1)$, the uniform distribution on $[0,1]$, is $T_{2}$-invariant
and ergodic.

Proposition 2.3 is a well-known fact in the theory of dynamical systems;
hence, we omit its proof, referring the interested reader to [10], for
example. In what follows, we will need a simple reformulation of this
statement for $r<1$.

######  Corollary 2.1.

$\mathcal{U}(r)$, the uniform distribution on $[0,r]$, is $t_{r}$-invariant
and ergodic.

###### Proof.

For $x\in[0,r]$, we have

(2.4) $t_{r}^{-1}(x)\ =\ r\cdot T_{2}^{-1}(x/r).$

To check that $\mathcal{U}(r)$ is $t_{r}$-invariant, we need to show that

(2.5) $\lvert t_{r}^{-1}(A)\rvert\cdot\frac{1}{r}\ =\ \lvert
A\rvert\cdot\frac{1}{r},$

for all $A\in\mathcal{B}([0,r])$, where $\lvert\cdot\rvert$ stands for the
Lebesgue measure on the line. Yet, by (2.4) and Proposition 2.3 (as
$\mathcal{U}(1)$ is $T_{2}$-invariant), we can write

$\lvert t_{r}^{-1}(A)\rvert\cdot r^{-1}\ =\ \lvert T_{2}^{-1}(A/r)\rvert\ =\
\lvert A/r\rvert\ =\ \lvert A\rvert\cdot r^{-1},$

which proves (2.5). Next, if $t_{r}$ was not ergodic, then we could find a set
$B\in\mathcal{B}([0,r])$ with $t_{r}^{-1}(B)=B$ and $\lvert B\rvert\in(0,r)$.
Consequently, by (2.4) we would obtain

$T_{2}^{-1}(B/r)\ =\ \frac{1}{r}\cdot t_{r}^{-1}(B)\ =\ B/r,$

while $B/r\in\mathcal{B}([0,1])$ and $\lvert B/r\rvert\in(0,1)$. This would
indicate that the map $T_{2}$ is not ergodic, contradicting Proposition 2.3. ∎

##  3\. Proof of uniqueness

Now we will show that $(\mu_{r},\nu_{r})$ is the only representation of
measure $m_{r}$. To this end, assume that
$m_{r}=\widetilde{\mu}_{r}+\widetilde{\nu}_{r}$ is some representation
$(\widetilde{\mu}_{r},\widetilde{\nu}_{r})\in\mathcal{R}$; we will demonstrate
that $\widetilde{\mu}_{r}=\mu_{r}$ and $\widetilde{\nu}_{r}=\nu_{r}$. From now
on, to simplify the notation, we will skip the index $r$ and write $\mu$,
$\nu$, $\widetilde{\mu}$, $\widetilde{\nu}$, $m$ instead of $\mu_{r}$,
$\nu_{r}$, $\widetilde{\mu}_{r}$, $\widetilde{\nu}_{r}$ and $m_{r},$
respectively. First, fix an arbitrary $A\in\mathcal{B}([0,r])$. By Definition
2.1, we have

$\int_{A}1\ \mathrm{d}\mu^{x}\ =\ \int_{A}x\
(\mathrm{d}\nu^{x}+\mathrm{d}\mu^{x})\ =\ \int_{A}x\ \mathrm{d}m^{x},$

and

$\int_{A}1\ \mathrm{d}\widetilde{\mu}^{x}\ =\ \int_{A}x\
(\mathrm{d}\widetilde{\nu}^{x}+\mathrm{d}\widetilde{\mu}^{x})\ =\ \int_{A}x\
\mathrm{d}m^{x},$

so $\mu^{x}(A)=\widetilde{\mu}^{x}(A)$. Similarly, we can deduce that
$\mu^{y}=\widetilde{\mu}^{y}$, and hence the marginal distributions of $\mu$
and $\widetilde{\mu}$ are equal. Because
$\mu+\nu=\widetilde{\mu}+\widetilde{\nu}$, we also get

(3.1) $\nu^{x}=\widetilde{\nu}^{x}\ \ \ \ \ \ \mathrm{and}\ \ \ \ \ \
\nu^{y}=\widetilde{\nu}^{y}.$

However, directly from (1.3), we infer that $\nu^{x}=\nu^{y}$ and both these
measures are proportional to the uniform distribution $\mathcal{U}(r)$. Let us
inspect the finite Borel signed measure $\delta=\widetilde{\nu}-\nu$. By the
Jordan decomposition theorem, we can find two nonnegative measures
$\delta_{+}$ and $\delta_{-}$, which are mutually singular and such that

(3.2) $\delta\ =\ \delta_{+}\ -\ \delta_{-}.$

As a consequence of (3.1), the marginal measures $\delta^{x}$ and $\delta^{y}$
are identically equal to zero. Therefore, using (3.2) we obtain

(3.3) $\delta^{x}_{+}=\delta^{x}_{-}\ \ \ \ \ \ \mathrm{and}\ \ \ \ \ \ \
\delta^{y}_{+}=\delta^{y}_{-}.$

The following observation is elementary, nonetheless very helpful.

######  Proposition 3.1.

We have $\delta_{-}\leq\nu$ and $\delta_{+}\leq\mu.$

###### Proof.

Since $\delta_{+}$ and $\delta_{-}$ are mutually singular, there exist two
disjoint Borel sets $S_{+},S_{-}$ with $S_{+}\cup S_{-}=[0,r]^{2}$, such that

(3.4) $\delta_{+}(A)\ =\ \delta_{+}(A\cap S_{+}),$ (3.5) $\delta_{-}(A)\ =\
\delta_{-}(A\cap S_{-}),$

for all Borel subsets $A\in\mathcal{B}([0,r]^{2})$. To show that
$\delta_{-}\leq\nu$, it is sufficient to check that

$\delta_{-}(B\cap S_{-})\ \leq\ \nu(B\cap S_{-}),$

for every choice of $B\in\mathcal{B}([0,r]^{2})$. By (3.2) and (3.4), we can
write

$-\delta_{-}(B\cap S_{-})\ =\ \delta(B\cap S_{-})\ =\ \widetilde{\nu}(B\cap
S_{-})-\nu(B\cap S_{-}),$

which yields

$\nu(B\cap S_{-})\ =\ \delta_{-}(B\cap S_{-})+\widetilde{\nu}(B\cap S_{-})\
\geq\ \delta_{-}(B\cap S_{-}).$

In the same way, by (3.2) and (3.5), we get

$\displaystyle\delta_{+}(B\cap S_{+})+(\widetilde{\mu}-\mu)(B\cap S_{+})$
$\displaystyle=\delta(B\cap S_{+})+(\widetilde{\mu}-\mu)(B\cap S_{+})$
$\displaystyle=(\widetilde{\nu}+\widetilde{\mu})(B\cap S_{+})-(\nu+\mu)(B\cap
S_{+})\ =\ 0,$

which results in

$\mu(B\cap S_{+})\ =\ \delta_{+}(B\cap S_{+})+\widetilde{\mu}(B\cap S_{+})\
\geq\ \delta_{+}(B\cap S_{+}),$

and proves that $\delta_{+}\leq\mu$. ∎

A central component of our argument is the next conclusion.

Figure 3. The preimage $t_{r}^{-1}(A)$ of a Borel set $A$ is made up of two
non-overlapping subsets with twice as small Lebesgue measure.

######  Proposition 3.2.

Assume that $\delta_{-}$ is not identically equal to zero and let

$\alpha_{-}\ =\ \frac{1}{\delta_{-}([0,r]^{2})}$

denote its norming constant. Then $\alpha_{-}\cdot\delta_{-}^{x}$ is a
$t_{r}$-invariant measure.

###### Proof.

Since $\delta_{-}$ is not a zero measure, we get

$\delta_{-}([0,r]^{2})\ =\ \delta_{-}^{x}([0,r])$

and hence $\alpha_{-}\cdot\delta_{-}^{x}$ is a probability measure. Now, by
Proposition 3.1 we have

$\delta_{-}\leq\nu\ \ \ \ \ \ \mathrm{and}\ \ \ \ \ \ \delta_{+}\leq\mu,$

which leads to

(3.6) $\mathrm{supp}(\delta_{-})\ \subset\ \mathrm{supp}(\nu)\ =\ \Gamma_{r}$

and

(3.7) $\mathrm{supp}(\delta_{+})\ \subset\ \mathrm{supp}(\mu)\ =\ \Delta_{r},$

respectively. Directly from (3.7), we see that $\mathrm{supp}(\delta_{+})$ is
a subset of the main diagonal, and therefore $\delta_{+}^{x}=\delta_{+}^{y}$.
Consequently, due to (3.3), we get the identities

(3.8) $\delta_{-}^{x}\ =\ \delta_{+}^{x}\ =\ \delta_{+}^{y}\ =\
\delta_{-}^{y}.$

Finally, take an arbitrary Borel set $A\in\mathcal{B}([0,r])$. From (3.6) we
know that $\mathrm{supp}(\delta_{-})$ is contained in $\Gamma_{r}$, i.e. in
the graph of function $t_{r}$, and so

(3.9) $\begin{split}\delta_{-}^{y}(A)&=\delta_{-}([0,r]\times
A)=\delta_{-}\Big{[}\big{(}[0,r]\times A\big{)}\cap\Gamma_{r}\Big{]}\\\ &=\
\delta_{-}^{x}\big{(}\\{x\in[0,r]:\ t_{r}(x)\in A\\}\big{)}\ =\
\delta_{-}^{x}\left(t_{r}^{-1}(A)\right),\end{split}$

see Figure 3. Combining (3.8) and (3.9), we conclude that

$\alpha_{-}\cdot\delta_{-}^{x}(A)\ \ =\ \ \alpha_{-}\cdot\delta_{-}^{y}(A)\ \
=\ \ \alpha_{-}\cdot\delta_{-}^{x}\left(t_{r}^{-1}(A)\right),$

which is the claim. ∎

We will require the following folklore remark. For the sake of completeness,
we also provide its straightforward proof.

######  Remark 3.1.

Take $s>0$ and let $T:[0,s]\rightarrow[0,s]$ be a Borel measurable
transformation. Let $\pi$ and $\sigma$ be two $T$-invariant probability
measures on $\mathcal{B}([0,s])$ and suppose that $\pi$ is additionally
ergodic. If $\sigma\ll\pi$ (i.e., if $\sigma$ is absolutely continuous with
respect to $\pi$), then $\sigma=\pi$.

###### Proof.

Let $\rho$ denote the Radon–Nikodym derivative of $\sigma$ with respect to
$\pi$: for any $C\in\mathcal{B}([0,s])$ we have $\sigma(C)=\int_{C}\rho\
\mathrm{d}\pi$. Let us distinguish the set $A=\\{\rho<1\\}$. Because $\pi$ and
$\sigma$ are $T$-invariant, we may write

(3.10) $\pi\left(A\setminus T^{-1}\left(A\right)\right)\ =\
\pi\left(T^{-1}\left(A\right)\setminus A\right),$

and

(3.11) $\sigma\left(A\setminus T^{-1}\left(A\right)\right)\ =\
\sigma\left(T^{-1}\left(A\right)\setminus A\right).$

Hence, if $\pi(A\setminus T^{-1}(A))>0$, then the very definition of $A$ we
get

(3.12) $\pi(A\setminus T^{-1}(A))\ >\ \int_{A\setminus T^{-1}(A)}\rho\
\mathrm{d}\pi\ =\ \sigma(A\setminus T^{-1}(A)).$

Analogously, we show that

(3.13) $\sigma\left(T^{-1}\left(A\right)\setminus A\right)\ =\
\int_{T^{-1}\left(A\right)\setminus A}\rho\ \mathrm{d}\pi\ \geq\
\pi\left(T^{-1}\left(A\right)\setminus A\right).$

Combining (3.11), (3.12) and (3.13), we obtain

$\pi\left(A\setminus T^{-1}\left(A\right)\right)\ >\
\pi\left(T^{-1}\left(A\right)\setminus A\right),$

which contradicts (3.10) and proves that $\pi(A\triangle T^{-1}(A))=0$. Since
measure $\pi$ is ergodic, we conclude that $\pi(A)\in\\{0,1\\}$. But
$\pi(\\{\rho<1\\})=1$ is impossible as $\pi$ and $\sigma$ are both
probabilistic. Therefore we must have $\pi(\\{\rho\geq 1\\})=1$. Again, since
$\pi$ and $\sigma$ are probabilistic, we deduce that $\rho=1$ $\pi$-almost
everywhere and thus $\sigma=\pi$. ∎

We proceed to the the primary objective of this section.

###### Proof of uniqueness.

Let us assume that $\delta_{-}$ is not identically equal to zero. Then by
Proposition 3.1 and equality (2.1), we know that

$\delta_{-}^{x}\leq\nu^{x}\ \ \ \ \ \ \mathrm{and}\ \ \ \ \ \
\nu^{x}=\frac{r}{c_{r}}\cdot\mathcal{U}(r),$

and hence $\alpha_{-}\cdot\delta_{-}^{x}\ \ll\mathcal{U}(r)$. Moreover, by
Proposition 3.2 and Corollary 2.1, we note that
$\alpha_{-}\cdot\delta_{-}^{x}$ and $\mathcal{U}(r)$ are both
$t_{r}$-invariant. But $\mathcal{U}(r)$ is also ergodic, so Remark 3.1 gives
$\alpha_{-}\cdot\delta_{-}^{x}=\mathcal{U}(r)$. Combining this with (3.3) and
Proposition 3.1, we get

(3.14) $\alpha_{-}^{-1}\cdot\mathcal{U}(r)\ =\ \delta_{-}^{x}\
=\delta_{+}^{x}\ \leq\ \mu^{x}.$

However, directly from (1.2), for $s\in(0,s_{0})$ we have

$\mu^{x}\left([0,s]\right)\ =\ c_{r}^{-1}\int_{0}^{s}\frac{z}{1-z}\
\mathrm{d}z\ <\ \alpha_{-}^{-1}\cdot s,$

provided $s_{0}$ is sufficiently small. This contradicts (3.14) and shows that
$\delta_{-}$ is a zero measure. Consequently, we get $\delta=\delta_{+}$:
$\delta$ is a nonnegative measure. But (3.1) gives $\delta^{x}=0$, so we must
have $\delta=0$ and $\widetilde{\nu}=\nu$. Finally, we write down the identity

$\widetilde{\mu}\ =\ m-\widetilde{\nu}\ =\ m-\nu\ =\ \mu,$

which gives $(\widetilde{\mu},\widetilde{\nu})=(\mu,\nu)$. This proves that
$(\mu,\nu)$ is a unique representation of $m$.∎

##  4\. Proof of minimality

It remains to verify the minimality of the representation $(\mu,\nu)$. Suppose
that

$\widetilde{\mu}\leq\mu\ \ \ \ \ \ \mathrm{and}\ \ \ \ \ \
\widetilde{\nu}\leq\nu,$

for some $(\widetilde{\mu},\widetilde{\nu})\in\mathcal{R}$. We need to show
that $(\widetilde{\mu},\widetilde{\nu})$ is proportional to $(\mu,\nu)$.

###### Proof of minimality.

Thanks to $\widetilde{\mu}\leq\mu$, we have

(4.1) $\mathrm{supp}(\widetilde{\mu})\ \subset\ \mathrm{supp}(\mu)\ =\
\Delta_{r}.$

In particular, $\mathrm{supp}(\widetilde{\mu})$ is a subset of the main
diagonal, which implies that

(4.2) $\widetilde{\mu}^{x}\ =\ \widetilde{\mu}^{y}.$

Next, using (4.2) and Definition 2.1, we get

(4.3) $\displaystyle\int_{A}x\ \mathrm{d}\widetilde{\nu}^{x}\ =$
$\displaystyle\ \int_{A}(1-x)\ \mathrm{d}\widetilde{\mu}^{x}$ $\displaystyle=$
$\displaystyle\ \int_{A}(1-x)\ \mathrm{d}\widetilde{\mu}^{y}\ =\ \int_{A}x\
\mathrm{d}\widetilde{\nu}^{y},$

for any Borel subset $A\in\mathcal{B}([0,r])$. Due to
$\widetilde{\nu}\leq\nu$, we have

$\widetilde{\nu}^{x}\ll\nu^{x}\ \ \ \ \ \ \mathrm{and}\ \ \ \ \ \
\widetilde{\nu}^{y}\ll\nu^{y}.$

Furthermore, based on (2.1), recall that $\nu^{x}=\nu^{y}$ and $\nu^{x}$ is
proportional to $\mathcal{U}(r)$. Applying the above observations, let
$\varphi_{x},\varphi_{y}$ denote the Radon–Nikodym derivatives of
$\widetilde{\nu}^{x}$ and $\widetilde{\nu}^{y}$ with respect to $\nu^{x}$. Put

$A_{+}=\left\\{\varphi_{x}>\varphi_{y}\right\\}\ \ \ \ \ \ \mathrm{and}\ \ \ \
\ \ A_{-}=\left\\{\varphi_{x}<\varphi_{y}\right\\}.$

Inserting sets $A_{+},A_{-}$ into (4.3), we obtain

(4.4) $\int_{A_{\pm}}x\left(\varphi_{x}-\varphi_{y}\right)\mathrm{d}\nu^{x}\
=\ \int_{A_{\pm}}x\ \mathrm{d}\widetilde{\nu}^{x}\ -\ \int_{A_{\pm}}x\
\mathrm{d}\widetilde{\nu}^{y}\ =\ 0,$

which shows that $A_{+}\cup A_{-}$ is a set of $\nu^{x}$-measure zero and
hence $\widetilde{\nu}^{x}=\widetilde{\nu}^{y}$. On the other hand, we have

(4.5) $\mathrm{supp}(\widetilde{\nu})\ \subset\ \mathrm{supp}(\nu)\ =\
\Gamma_{r},$

so $\mathrm{supp}(\widetilde{\nu})$ is a subset of the graph of the function
$t_{r}$. Thus, we can simply repeat the reasoning from the proof of
Proposition 3.2 – rewriting the equation (3.9) with $\delta_{-}$ replaced by
$\widetilde{\nu}$, we get that $\widetilde{\nu}$ is a $t_{r}$-invariant
measure (up to proportionality). Then, again by Remark 3.1,

(4.6) $\widetilde{\nu}^{x}\ =\ \alpha\cdot\mathcal{U}(r)\ =\
\alpha^{\prime}\cdot\nu^{x},$

for some factors $\alpha,\alpha^{\prime}\geq 0$. Combining (4.6) with
Definition 2.1, we get

(4.7) $\displaystyle\int_{A}(1-x)\ \mathrm{d}\widetilde{\mu}^{x}\ =$
$\displaystyle\ \int_{A}x\ \mathrm{d}\widetilde{\nu}^{x}$ $\displaystyle=$
$\displaystyle\ \alpha^{\prime}\cdot\int_{A}x\ \mathrm{d}\nu^{x}\ =\
\alpha^{\prime}\cdot\int_{A}(1-x)\ \mathrm{d}\mu^{x},$

for any Borel subset $A\in\mathcal{B}([0,r])$. Just as in (4.4), plugging the
Radon–Nikodym derivative of $\widetilde{\mu}^{x}$ with respect to $\mu^{x}$
into (4.7), we check that $\widetilde{\mu}^{x}=\alpha^{\prime}\cdot\mu^{x}$.
Altogether with (4.6), (4.1) and (4.5), this yields
$(\widetilde{\mu},\widetilde{\nu})=\alpha^{\prime}\cdot(\mu,\nu)$. The proof
of the minimality is hence complete. ∎

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