# Joint Semantic Communication and Target Sensing for 6G Communication System

Yinchao Yang1, Mohammad Shikh-Bahaei1, Zhaohui Yang23, Chongwen Huang23, Wei
Xu4,
and Zhaoyang Zhang23  ${1}$Department of Engineering King’s College London
London, UK
${2}$College of Information Science and Electronic Engineering, Zhejiang
University, Hangzhou, China
${3}$Zhejiang Provincial Key Laboratory of Info. Proc., Commun. & Netw.
(IPCAN), Hangzhou, China
${4}$National Mobile Communications Research Laboratory, Southeast University,
Nanjing, China
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper investigates the secure resource allocation for a downlink
integrated sensing and communication system with multiple legal users and
potential eavesdroppers. In the considered model, the base station (BS)
simultaneously transmits sensing and communication signals through beamforming
design, where the sensing signals can be viewed as artificial noise to enhance
the security of communication signals. To further enhance the security in the
semantic layer, the semantic information is extracted from the original
information before transmission. The user side can only successfully recover
the received information with the help of the knowledge base shared with the
BS, which is stored in advance. Our aim is to maximize the sum semantic
secrecy rate of all users while maintaining the minimum quality of service for
each user and guaranteeing overall sensing performance. To solve this sum
semantic secrecy rate maximization problem, an iterative algorithm is proposed
using the alternating optimization method. The simulation results demonstrate
the superiority of the proposed algorithm in terms of secure semantic
communication and reliable detection.

###### Index Terms:

Integrated Sensing and Communication, Transmit Beamforming, Semantic
Communication.

##  I Introduction

The bottleneck of spectrum shortage is brought on by the proliferation of
emerging applications with high data-rate requirements, such as virtual
reality, holographic communication, and high-mobility sensing [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11]. To meet the strict requirements in emerging applications,
three functions must be synchronized, i.e., mobile sensing, communication, and
computation. While these three functions operate independently in conventional
wireless networks, they cannot meet the demands of the sixth-generation (6G)
communication system including immersive communication, hyper-reliable and low
latency communications, integrated sensing and communication, and integrated
AI and communication.

It is therefore imperative that academics and industry work together to
develop novel approaches to this problem. A rising technology, integrated
sensing and communication (ISAC), has recently gained attention. Researchers
initially focused on sharing the spectrum between communication and sensing.
It was called communication and radar spectrum sharing [12]. Thereafter,
further designs have been proposed where the spectrum and hardware
infrastructure are both shared between communication and sensing [13]. As a
result of its dual functions, ISAC significantly increases spectrum efficiency
as well as supports emerging applications, which will play an important role
in future networks.

With the rapid development of artificial intelligence, semantic communication
has also proven to be an effective technology for addressing resource scarcity
[14, 15, 16, 17]. A breakthrough in semantic communication is that it
transcends Shannon’s paradigm, as opposed to conventional communication, which
is constrained by Shannon’s capacity limit. Rather than transmitting the
entire message, semantic communication focuses on extracting and transmitting
just its meaning. Several semantic coding strategies have been proposed for
different data formats. For example, the work in [18] proposed machine
learning approaches for processing text messages. Moreover, the authors in
[19, 20] concentrated on machine learning applications for image semantic
coding.

To achieve efficient transmission, semantic communication works in the
following steps. In the first step, the transmitter extracts the meaning of
the source messages (e.g., text, image, video) using its own knowledge base.
The transmitter’s knowledge base is built on common knowledge (shared between
both the transmitter and the receiver) and private knowledge. With the help of
autoencoders [21], this step can be achieved. In the second step, semantic
messages are then transmitted over wireless channels. In the third step, the
receiver recovers the meaning of the original messages by utilizing its local
knowledge base. However, the above semantic communication works [18, 19, 20,
21] did not consider the integration of sensing and semantic communication,
even though the joint design of sensing and semantic resource can further
enhance the performance of the system.

In this paper, we investigate the integrated sensing and semantic
communication (ISSC) model, which embeds semantic communication into the ISAC
model, not only making more spectrum resources available but also adding an
extra layer of security. The ISSC model differs from traditional physical
layer security [22], in that, even if an eavesdropper intercepts a message, he
or she may not be able to understand its meaning in the semantic layer. To the
best of our knowledge, there is no existing work that focuses on joint
semantic communication and sensing. The main contributions of this paper are
as follows:

  1. 1.

We propose the framework of an integrated sensing and semantic communication
system as a spectrum-efficient and secure method. To ensure the security of
the system, the sensing signal can be viewed as artificial noise for the
communication signal and the semantic communication technique is used to
protect the meaning of the transmitted communication signal.

  2. 2.

In order to determine the explicit performance indicator of semantic
communication, we take into account the well-known BLEU score. The semantic
rate expression is obtained through considering the effect of both semantic
compression ratio and semantic computation power. Based on the considered
model, the joint sensing, communication, and semantic computation optimization
problem is formulated by maximizing the sum secure semantic rate. To solve
this problem, an alternating algorithm is proposed with iteratively solving
the sensing beamforming, communication beamforming, and semantic compression
ratio.

##  II System Model

Figure 1: An ISSC system model with $L$ passive targets and $K$ communication
users.

As illustrated in Fig. 1, we consider an ISSC system with a BS equipped with a
uniform linear array (ULA) of $N$ antennas. The antennas are co-located for
target detection and downlink semantic communication. The BS communicates with
$K$ cellular users (CUs), and each CU $k\in\mathcal{K}$ is equipped with a
single antenna. Meanwhile, the BS aims to identify $L$ point-like targets, and
each target $l\in\mathcal{L}$ is potentially a passive eavesdropper.

The received signal at the $k$-th CU can be characterized as follows:

$y_{k}=\mathbf{h}_{k}^{H}\mathbf{x}+n_{k},$ (1)

where $\mathbf{h}_{k}\in\mathbb{C}^{N}$ is the channel vector for the $k$-th
CU, $\mathbf{x}\in\mathbb{C}^{N}$ is the transmitted signal, and
$n_{k}\sim\mathcal{CN}(0,\sigma^{2}_{c})$ is the communication noise for the
$k$-th CU. On the target side, the received signal can be formulated by

$y_{l}=\alpha_{l}\mathbf{a}^{H}(\theta_{l})\mathbf{x}+n_{l}=\mathbf{h}_{l}^{H}\mathbf{x}+n_{l},$
(2)

where $\alpha_{l}$ and $n_{l}\sim\mathcal{CN}(0,\sigma^{2}_{r})$ are the path-
loss coefficient and the sensing noise for the $l$-th target, respectively.
The steering vector is denoted by $\mathbf{a}(\theta_{l})\in\mathbb{C}^{N}$
with $\theta_{l}$ being the direction of the $l$-th target,
$|\theta_{l}|\leq\frac{\pi}{2},\forall l$, which can be formulated by

$\mathbf{a}^{H}(\theta_{l})=\begin{bmatrix}1&e^{j2\pi\frac{d}{\lambda}\sin(\theta_{l})}&\cdots&e^{j2\pi(N-1)\frac{d}{\lambda}\sin(\theta_{l})}\end{bmatrix}.$
(3)

In (3), $\lambda$ is the wavelength and $d$ is the distance between any two
adjacent antennas. The echo signal received at the BS can be formulated by

$\mathbf{\hat{y}}_{l}=\beta_{l}\mathbf{a}(\theta_{l})\mathbf{a}^{H}(\theta_{l})\mathbf{x}+\mathbf{n}_{l},$
(4)

where $\beta_{l}$ and $\mathbf{n}_{l}\in\mathbb{C}^{N}$ are the path loss
coefficient and noise vector, respectively.

In the considered ISSC system, both semantic communication signals and sensing
signals are simultaneously transmitted. Thus, the transmitted signal
$\mathbf{x}$ at the BS can be formulated as

$\mathbf{x}=\mathbf{W}\mathbf{c}+\mathbf{R}\mathbf{z},$ (5)

where $\mathbf{W}\in\mathbb{C}^{N\times K}$ denotes the precoding matrix and
$\mathbf{c}=[c_{1},c_{2},c_{k},\dotsc,c_{K}]\in\mathbb{C}^{K}$ denotes the
semantic message. The semantic message $c_{k}$ is generated from the
conventional communication message $m_{k}$, i.e., $c_{k}=f(m_{k})$. Here,
$f(\cdot)$ is a function that maps a conventional communication message to a
semantic message. For instance, $f(\cdot)$ means a neural network with
embedded large language model[18], where the semantic message is the output of
the neural network. After that, the semantic message is encoded via channel
coding before transmission. Another example is to consider $f(\cdot)$ as an
autoencoder that achieves joint source channel coding[18]. In this paper, we
assume that the semantic message is extracted via an autoencoder.

Moreover, in (5), $\mathbf{R}\in\mathbb{C}^{N\times L}$ represents the radar
beamforming matrix and
$\mathbf{z}=[z_{1},z_{2},z_{l},\dotsc,z_{L}]\in\mathbb{C}^{L}$ is the sensing
signal. Without the loss of generality, we make the following assumptions:

  1. i).

There is no correlation between the confidential message and the radar signal
[23]. As a result, we have

$\mathbb{E}(\mathbf{c}(\mathbf{z})^{H})=\mathbf{0}_{K\times L}.$ (6)

  2. ii).

There is no correlation between the confidential messages for different CUs.
Specifically, this assumption can be stated as follows:

$\mathbb{E}(\mathbf{c}(\mathbf{c})^{H})=\mathbf{I}_{K}.$ (7)

  3. iii).

There is no correlation between the sensing signals for different targets. In
this case, the assumption can be expressed as follows:

$\mathbb{E}(\mathbf{z}(\mathbf{z})^{H})=\mathbf{I}_{L}.$ (8)

As a result, the covariance matrix of the transmit waveform can be derived as
follows:

$\mathbf{R}_{x}=\mathbb{E}[\mathbf{x}\mathbf{x}^{H}]=\sum_{k=1}^{K}\mathbf{W}_{k}+\sum_{l=1}^{L}\mathbf{R}_{l},$
(9)

where $\mathbf{W}_{k}\equiv\mathbf{w}_{k}\mathbf{w}_{k}^{H}$ and
$\mathbf{R}_{l}\equiv\mathbf{r}_{l}\mathbf{r}_{l}^{H}$. Hence, as given in
[24], the beampattern can be described by :

$p(\phi)=\mathbf{a}^{H}(\phi)\mathbf{R}_{x}\mathbf{a}(\phi),$ (10)

where $|\phi|\leq\frac{\pi}{2}$ represents a generic angle covering a fine
grid of angles. In this regard, optimizing the transmit beampattern is
equivalent to optimizing the covariance matrix.

###  II-A Semantic Communication

We define the semantic transmission rate as the number of bits received by the
users after de-extracting the semantic information, hence the formulation is
given by

$S_{k}=\frac{1}{\rho_{k}}\log_{2}(1+\gamma_{k}),$ (11)

where $\rho_{k}=\frac{\text{len}(c_{k})}{\text{len}(m_{k})},0\leq\rho_{k}\leq
1$ is the semantic extraction ratio and $\text{len}(\cdot)$ means the number
of bits of a message. $\gamma_{k}$ is SINR for the $k$-th CU which is given by

$\footnotesize\gamma_{k}=\frac{\operatorname{Tr}(\mathbf{h}_{k}\mathbf{h}_{k}^{H}\mathbf{W}_{k})}{\operatorname{Tr}(\mathbf{h}_{k}\mathbf{h}_{k}^{H}\sum^{K}_{k^{\prime}=1,k^{\prime}\neq
k}\mathbf{W}_{k^{\prime}})+\operatorname{Tr}(\mathbf{h}_{k}\mathbf{h}_{k}^{H}\sum_{l=1}^{L}\mathbf{R}_{l})+\sigma^{2}_{c}}.$
(12)

Furthermore, in semantic communication, it is important to consider the
similarity between the original message $m_{k}$ and the extracted message
$c_{k}$. The BLEU score, as defined in [25], measures the text message
similarity, whose function is given as follows:

$\text{BP}=\begin{cases}1,&\text{if }\text{len}(c)>\text{len}(m),\\\
e^{1-\frac{\text{len}(m)}{\text{len}(c)}},&\text{if
}\text{len}(c)\leq\text{len}(m).\end{cases}$
$\text{BLEU}=\text{BP}\exp(\sum_{g=1}^{G}w_{g}\log p_{g}),$

where $w_{g}$ defines the weight of the g-grams and $p_{g}$ is the g-grams
precision score. In our model, the length of $c$ is always less than or equal
to the length of $m$. Together with the definition of
$\rho_{k}=\frac{\text{len}(c_{k})}{\text{len}(m_{k})}$, we can rewrite the
BLEU score for the $k$-th CU as:

$\text{BLEU}_{k}=e^{1-\frac{1}{\rho_{k}}}\exp(\sum_{g=1}^{G}w_{g,k}\log
p_{g,k}).$ (13)

When designing the system, we need to fix the value of $p_{g,k}$ and the lower
bound of $\text{BLEU}_{k}$. The reasons are

  1. i.

When optimizing the extraction ratio $\rho_{k}$, we need to make sure that
each user can accurately decode the message.

  2. ii.

When $\text{BLEU}_{k}$ is extremely low, it means the extraction workload for
the BS is huge, hence leading to a very high computation cost.

######  Lemma 1.

By choosing an appropriate lower bound value for all $\text{BLEU}_{k}$ and
appropriate values of $p_{g,k}$ for each CU, we can obtain the lower bound of
$\rho_{k}$ as:

$\rho_{k}\geq\frac{1}{1-\ln Q+\sum_{g=1}^{G}w_{g,k}\log p_{g,k}}.$ (14)

In (14), $Q$ is the lower bound of the all $\text{BLEU}_{k}$ and $\ln
Q\leq\sum_{g=1}^{G}w_{g,k}\log p_{g,k}$. Additionally, the values of $w_{g,k}$
and $p_{g,k}$ depend on each CU.

###### Proof.

See Appendix A ∎

###  II-B Semantic Security

To measure the level of security, we define the semantic secrecy rate (SSR) of
user $k$ as the difference between the semantic transmission rate of user $k$
and the most robust semantic transmission rate of the eavesdropper. To give
the formulation, we first need to define the SNR between the BS and $l$-th
eavesdropper about the confidential message for the $k$-th CU:

$\Gamma_{l|k}=\frac{\operatorname{Tr}(\mathbf{h}_{l}\mathbf{h}_{l}^{H}\mathbf{W}_{k})}{\operatorname{Tr}(\mathbf{h}_{l}\mathbf{h}_{l}^{H}\sum_{l=1}^{L}\mathbf{R}_{l})+\sigma^{2}_{r}}.$
(15)

Hence, the semantic transmission rate of the $l$-th eavesdropper related to
the $k$-th CU is given by

$S_{l|k}=\frac{1}{\rho_{k}}\log_{2}(1+\Gamma_{l|k}).$ (16)

In this way, the worst-case SSR of the $k$-th CU among all possible
eavesdroppers can be formulated by

$SSR_{k}=\min_{l\in L}[S_{k}-S_{l|k}]^{+},$ (17)

where $[v]^{+}$ means $\max(0,v)$.

###  II-C Power Budget

Extracting semantic information from a conventional message is heavily based
on machine learning methods. Therefore, we must consider the computation power
in the overall power budget. Following the results from [26, 27], the
segmented semantic extraction ratio has an exponential relationship with the
computation power: when the semantic extraction ratio is closer to zero, the
gradient of the function is steeper. However, the function proposed in [27]
depends on many hyper-parameters, such as the interval and the gradient of
each segment. To avoid manually defining all the hyper-parameters, we instead
use a natural logarithm function to calculate the computation power:

$P_{\text{comp}}=\sum_{k=1}^{K}F\ln(1/\rho_{k}),$ (18)

where F is a coefficient that converts a magnitude to its power. On the other
hand, the communication and sensing energy consumption is given by

$\operatorname{Tr}(\sum_{k=1}^{K}\mathbf{W}_{k}+\sum_{l=1}^{L}\mathbf{R}_{l})=P_{\text{c\&s}}.$
(19)

Additionally, the overall power consumption is limited to the power budget:

$P_{\text{comp}}+P_{\text{c\&s}}\leq P_{t},$ (20)

with $P_{t}$ being the power budget.

###  II-D Sensing Performance

During the design of a sensing-only system, we obtain the radar beamforming
matrix $\mathbf{R}_{d}$, which represents the optimal beam pattern for sensing
only. This design can be formulated by maximizing the beams directed toward
the targets. As part of this optimization problem, it is necessary to make
sure there is no cross-correlation between different targets while also
guaranteeing the transmit power budget. Therefore, the maximization problem is
given by

$\displaystyle\max_{t,\mathbf{R}_{d}}\quad$ $\displaystyle t$ (21a) s.t.
$\displaystyle\mathbf{a}^{H}(\theta_{l})\mathbf{R}_{d}\mathbf{a}(\theta_{l})-\mathbf{a}^{H}(\theta_{m})\mathbf{R}_{d}\mathbf{a}(\theta_{m})\geq
t,\forall\theta_{l},\forall\theta_{m},$ (21b)
$\displaystyle\mathbf{a}^{H}(\theta_{l})\mathbf{R}_{d}\mathbf{a}(\theta_{l^{\prime}})=0,\quad
l^{\prime}\neq l,\quad\forall l,$ (21c)
$\displaystyle\operatorname{Tr}(\mathbf{R}_{d})\leq P_{t},$ (21d)
$\displaystyle\mathbf{R}_{d}\succeq 0,\mathbf{R}_{d}=\mathbf{R}_{d}^{H},$
(21e)

where $t$ is a threshold value that we wish to maximize, $\theta_{l}$
represents the angle of the $l^{th}$ target and $\theta_{m}\in\Omega$
represents an angle in the sidelobe region. Optimization problem (21) in a
pure sensing system is a standard convex optimization problem and thus can be
efficiently solved by numerical tools.

In order to guarantee the sensing performance of the ISSC system, we must
guarantee that the mismatch between
$\sum_{k=1}^{K}\mathbf{W}_{k}+\sum_{l=1}^{L}\mathbf{R}_{l}$ and
$\mathbf{R}_{d}$ should be less than a pre-defined threshold $\xi$, i.e.,

$|\mathbf{R}_{d}-(\sum_{k=1}^{K}\mathbf{W}_{k}+\sum_{l=1}^{L}\mathbf{R}_{l})|^{2}\leq\xi.$
(22)

##  III Beamforming design for Integrated sensing and semantic communication

In this section, we assume the channel of all CUs and the locations of all
targets are available at the BS.

###  III-A Problem Formulation

To design secure ISSC beamforming and to optimize the semantic extraction
ratio, our goal is to maximize the sum semantic secrecy rate. The constraints
of this optimization problem include the boundary of the extraction ratio
$\rho_{k}$ as given in (14). Next, to guarantee the minimum quality of service
for each CU, we require the semantic transmission rate for each CU to be above
a pre-defined value $\varsigma$. Additionally, the transmit power budget as
defined in (20), the beampattern mismatch as given in (21), and the matrix
positive semi-definite property need to be satisfied. Finally, to ensure the
achievable semantic transmission rate can be attained by using single stream
transmit beamforming for each CU, a rank-one constraint is added. Note that if
the rank-one constraint is not in place, more sophisticated transceiver
schemes are needed [28].

As a result, the optimization problem can be formulated as follows:

$\displaystyle\max_{\mathbf{W}_{k},\mathbf{R}_{l},\rho_{k}}\;$
$\displaystyle\sum_{k=1}^{K}SSR_{k}$ (23a) s.t. $\displaystyle\frac{1}{1-\ln
Q+\sum_{g=1}^{G}w_{g,k}\log p_{g,k}}\leq\rho_{k}\leq 1,\forall k,$ (23b)
$\displaystyle S_{k}\geq\varsigma,\forall k,$ (23c) $\displaystyle
P_{\text{comp}}+P_{\text{c\&s}}\leq P_{t},$ (23d)
$\displaystyle|\mathbf{R}_{d}-(\sum_{k=1}^{K}\mathbf{W}_{k}+\sum_{l=1}^{L}\mathbf{R}_{l})|^{2}\leq\xi,$
(23e) $\displaystyle\mathbf{W}_{k}\succeq
0,\mathbf{W}_{k}=\mathbf{W}_{k}^{H},\quad\forall k,$ (23f)
$\displaystyle\mathbf{R}_{l}\succeq
0,\mathbf{R}_{l}=\mathbf{R}_{l}^{H},\forall l,$ (23g)
$\displaystyle\text{rank}(\mathbf{W}_{k})=1,\quad\forall k.$ (23h)

The worst-case SSR in (23a) can be modified to:

$SSR_{k}=[S_{k}-\max_{l\in L}S_{l|k}]^{+}.$ (24)

###  III-B Algorithm Design

Through introducing auxiliary variables $\lambda_{k},\forall k$, together with
(24), we can transform the first part of (23a) into the following form:

$\max_{\mathbf{W}_{k},\mathbf{R}_{l},\rho_{k},\lambda_{k}}\sum_{k=1}^{K}(S_{k}-\frac{1}{\rho_{k}}\log_{2}(1+\lambda_{k})),$
(25)

with an additional constraint of

$\Gamma_{l|k}\leq\lambda_{k},\forall k,\forall l.$ (26)

However, (25) is a non-concave function due to the logarithm terms in the
equation. We can apply first-order Taylor expansion on $S_{k}$, which leads
to:

$\displaystyle S_{k}$ (27)
$\displaystyle=\frac{1}{\rho_{k}}\log_{2}(\frac{\mathbf{h}_{k}^{H}\sum_{k=1}^{K}\mathbf{W}_{k}\mathbf{h}_{k}+\mathbf{h}_{k}^{H}\sum_{l=1}^{L}\mathbf{R}_{l}\mathbf{h}_{k}+\sigma_{c}^{2}}{\mathbf{h}_{k}^{H}\sum_{k^{\prime}=1,k^{\prime}\neq
k}^{K}\mathbf{W}_{k^{\prime}}\mathbf{h}_{k}+\mathbf{h}_{k}^{H}\sum_{l=1}^{L}\mathbf{R}_{l}\mathbf{h}_{k}+\sigma_{c}^{2}})$
$\displaystyle=\frac{1}{\rho_{k}}\Big{(}\log_{2}(A_{k})-\log_{2}(B_{k})\Big{)}$
$\displaystyle=\frac{1}{\rho_{k}}\Big{(}\log_{2}(A_{k})-\log_{2}(B_{k}^{i})-\frac{1}{B_{k}^{i}\ln(2)}(B_{k}-B_{k}^{i})\Big{)}.$

As a result, the maximization of the concave expression $S_{k}$ in (27) falls
in the convex optimization.

Similarly, by applying first-order Taylor expansion on
$-\frac{1}{\rho_{k}}\log_{2}(1+\lambda_{k})$, we can further have:

$\displaystyle-\frac{1}{\rho_{k}}\Big{(}\log_{2}(1+\lambda_{k})\Big{)}=-\frac{1}{\rho_{k}}\Big{(}\log_{2}(C_{k})\Big{)}$
(28)
$\displaystyle=-\frac{1}{\rho_{k}}\Big{(}\log_{2}(C_{k}^{i})+\frac{1}{C_{k,l}^{i}\ln(2)}(C_{k}-C_{k}^{i})\Big{)}.$

In (27) and (28), $A_{k}$, $B_{k}$, $C_{k}$ are variables and $B_{k}^{i}$,
$C_{k}^{i}$ stands for the value of $B_{k}$, $C_{k}$ in the $i$-th iteration,
which can be calculated by

$\displaystyle B_{k}^{i}$
$\displaystyle=\mathbf{h}_{k}^{H}\sum_{k^{\prime}=1,k^{\prime}\neq
k}^{K}\mathbf{W}^{i}_{k^{\prime}}\mathbf{h}_{k}+\mathbf{h}_{k}^{H}\sum_{l=1}^{L}\mathbf{R}^{i}_{l}\mathbf{h}_{k}+\sigma_{c}^{2}$
(29) $\displaystyle C_{k}^{i}$ $\displaystyle=1+\lambda^{i}_{k},$

where $W^{i}_{k^{\prime}}$, $R_{l}^{i}$ and $\lambda^{i}_{k}$ are the values
of $W_{k^{\prime}}$, $R_{l}$ and $\lambda_{k}$ in the $i$-th iteration.

Furthermore, by dropping the rank-one constraint using SDR, problem (23) can
be transformed into the following form

$\displaystyle\max_{\psi}$
$\displaystyle\sum_{k=1}^{K}\frac{1}{\rho_{k}}\Big{(}\log_{2}(A_{k})-\log_{2}(B_{k}^{i})-\frac{1}{B_{k}^{i}\ln(2)}(B_{k}-B_{k}^{i})$
$\displaystyle-\log_{2}(C_{k}^{i})-\frac{1}{C_{k}^{i}\ln(2)}(C_{k}-C_{k}^{i})\Big{)},$
(30a) s.t. $\displaystyle\frac{1}{1-\ln Q+\sum_{g=1}^{G}w_{g,k}\log
p_{g,k}}\leq\rho_{k}\leq 1,\forall k,$ (30b)
$\displaystyle\frac{1}{\rho_{k}}\Big{(}\log_{2}(A_{k})-\log_{2}(B_{k}^{i})-\frac{1}{B_{k}^{i}\ln(2)}$
$\displaystyle(B_{k}-B_{k}^{i})\Big{)}\geq\varsigma,\forall k,$ (30c)
$\displaystyle\Gamma_{l|k}\leq\lambda_{k},\forall k,\forall l,$ (30d)
$\displaystyle P_{\text{comp}}+P_{\text{c\&s}}\leq P_{t},$ (30e)
$\displaystyle|\mathbf{R}_{d}-(\sum_{k=1}^{K}\mathbf{W}_{k}+\sum_{l=1}^{L}\mathbf{R}_{l})|^{2}\leq\xi,$
(30f) $\displaystyle\mathbf{W}_{k}\succeq
0,\mathbf{W}_{k}=\mathbf{W}_{k}^{H},\forall k,$ (30g)
$\displaystyle\mathbf{R}_{l}\succeq
0,\mathbf{R}_{l}=\mathbf{R}_{l}^{H},\forall l,$ (30h)

where $\psi=[\mathbf{W}_{k},\mathbf{R}_{l},\rho_{k},\lambda_{k}]$. To solve
(30), we consider using alternating optimization:

####  III-B1 Step 1

With given $\rho_{k}$ and $\lambda_{k}$ in problem (30), the joint sensing and
communication beamforming optimization problem can be given by

$\displaystyle\max_{\mathbf{W}_{k},\mathbf{R}_{l}}\;$
$\displaystyle\sum_{k=1}^{K}\frac{1}{\rho_{k}}\Big{(}\log_{2}(A_{k})-\log_{2}(B_{k}^{i})-\frac{1}{B_{k}^{i}\ln(2)}$
$\displaystyle(B_{k}-B_{k}^{i})-\log_{2}(1+\lambda_{k})\Big{)}$ (31a) s.t.
$\displaystyle\frac{1}{\rho_{k}}\Big{(}\log_{2}(A_{k})-\log_{2}(B_{k}^{i})-\frac{1}{B_{k}^{i}\ln(2)}$
$\displaystyle(B_{k}-B_{k}^{i})\Big{)}\geq\varsigma,\forall k,$ (31b)
$\displaystyle\operatorname{Tr}(\mathbf{h}_{l}\mathbf{h}_{l}^{H}\mathbf{W}_{k})-$
$\displaystyle\lambda_{k}(\operatorname{Tr}(\mathbf{h}_{l}\mathbf{h}_{l}^{H}\sum_{l=1}^{L}\mathbf{R}_{l})+\sigma^{2}_{r})\leq
0,\forall k,\forall l,$ (31c) $\displaystyle
P_{\text{comp}}+P_{\text{c\&s}}\leq P_{t}$ (31d)
$\displaystyle|\mathbf{R}_{d}-(\sum_{k=1}^{K}\mathbf{W}_{k}+\sum_{l=1}^{L}\mathbf{R}_{l})|^{2}\leq\xi,$
(31e) $\displaystyle\mathbf{W}_{k}\succeq
0,\mathbf{W}_{k}=\mathbf{W}_{k}^{H},\quad\forall k$ (31f)
$\displaystyle\mathbf{R}_{l}\succeq
0,\mathbf{R}_{l}=\mathbf{R}_{l}^{H},\forall l,$ (31g)

which is convex and can be effectively solved via the standard convex
optimization tool.

####  III-B2 Step 2

With fixed $\rho_{k}$ as well as the obtained values of $\mathbf{W}_{k}$ and
$\mathbf{R}_{k}$ from Step 1, the optimization of $\lambda_{k}$ is

$\displaystyle\max_{\lambda_{k}}\;$
$\displaystyle\sum_{k=1}^{K}\frac{1}{\rho_{k}}\Big{(}\log_{2}(A_{k})-\log_{2}(B_{k})$
$\displaystyle-\log_{2}(C_{k}^{i})-\frac{1}{C_{k}^{i}\ln(2)}(C_{k}-C_{k}^{i})\Big{)}$
(32a) s.t. $\displaystyle\lambda_{k}\geq\Gamma_{l|k},\forall k,\forall l,$
(32b)

where $A_{k}$, $B_{k}$ and $\Gamma_{l|k}$ are known values. $C_{k}^{i}$ is
updated iteratively and $C_{k}=1+\lambda_{k}$ is the variable. Problem (32) is
convex and can be effectively solved.

####  III-B3 Step 3

With given $\mathbf{W}_{k}$, $\mathbf{R}_{k}$, $\lambda_{k}$, the optimization
of semantic compression ratio $\rho_{k}$ is

$\displaystyle\max_{\rho_{k}}$
$\displaystyle\sum_{k=1}^{K}\frac{1}{\rho_{k}}\Big{(}\log_{2}(A_{k})-\log_{2}(B_{k})-\log_{2}(C_{k})\Big{)}$
(33a) s.t. $\displaystyle\frac{1}{1-\ln Q+\sum_{g=1}^{G}w_{g,k}\log
p_{g,k}}\leq\rho_{k}\leq 1,\forall k$ (33b)
$\displaystyle\frac{1}{\rho_{k}}\Big{(}\log_{2}(A_{k})-\log_{2}(B_{k})\Big{)}\geq\varsigma,\forall
k$ (33c) $\displaystyle P_{\text{comp}}\leq P_{t}-P_{\text{c\&s}}$ (33d)

which is also convex and can be solved through the dual method.

####  III-B4 Step 4

As a last step, we use Gaussian randomisation to recover rank-one solutions.

The detailed procedures for solving problem (30) are shown in Algorithm 1.

Algorithm 1 Iterative Sensing, Communication, and Semantic Optimization
Algorithm

1: Initialize $\rho_{k}$, $\mathbf{W}_{k}^{i}$, $\mathbf{R}_{k}^{i}$,
$\lambda_{k}$, $B_{k}^{i}$ and $C_{k}^{i}$.

2: repeat

3: repeat

4: Solve (31).

5: Update $\mathbf{W}_{k}^{i+1}$ and $\mathbf{R}_{k}^{i+1}$.

6: until $|\mathbf{W}^{i+1}-\mathbf{W}^{i}|\leq\varrho_{1}$,
$|\mathbf{R}^{i+1}-\mathbf{R}^{i}|\leq\varrho_{2}$.

7: repeat

8: Solve (32).

9: Update $C_{k}^{i+1}$.

10: until $|C_{k}^{i+1}-C_{k}^{i}|\leq\varrho_{3}$.

11: Solve (33) and update $\rho_{k}$.

12: until converge

13: Apply Gaussian randomisation.

##  IV Numerical results

During our simulation, we assume the BS is equipped with ULA that has half-
wavelength spacing. The number of antennas is 18. The locations of targets are
at $[-35^{\circ},5^{\circ},45^{\circ}]$ and the locations of CUs are at
$[-30^{\circ},20^{\circ}]$. Path loss coefficients are generated randomly
between $0.001$ and $0.01$. The value of $\varsigma$ is equal to 1 and $\xi$
is equal to 5. The average noise power is -60 dBm. The value of $F$ in (18) is
10. The power budget varies from 5 dBm to 25 dBm. For comparison, the first
benchmark we use is by setting $\rho_{k}=1,\forall k$ and never updating it.
This means there is no semantic extraction. The second benchmark is [29],
where the SINR threshold upper bound is 25.

Figure 2: Worst-case semantic secrecy rate for different users.

Fig. 2 shows that when semantic extraction is involved in the system, the
worst-case SSR improves significantly compared to benchmark 1 and benchmark 2.
Furthermore, user 1 achieves a slightly lower worst-case SSR than user 2. This
is because we assume user 1 is less capable of recovering the semantic
message. Therefore, the lower bound of $\rho_{k=1}$ is higher than that of
$\rho_{k=2}$, which leads to a lower worst-case SSR for user 1. In Fig. 3, we
evaluate the sensing performance using the Multiple SIgnal Classification
(MUSIC) spectrum. In both our algorithm and the benchmarks, we are able to
detect peaks at the locations of targets, which indicates that the radar is
capable of detecting all targets accurately. Thus, our algorithm guarantees
the same sensing performance as the benchmarks while providing higher
security.

Figure 3: MUSIC Spectrum

##  V Conclusion

This paper provided a unified framework of integrated sensing and semantic
communication. A joint design of transmit beamforming and semantic extraction
ratio was presented. By using alternating optimization, we proposed an
iterative algorithm to optimize the sum of the semantic secrecy rate for all
users. Based on our simulation results, we found that our design provides
higher security than the existing ISAC frameworks while guaranteeing the same
sensing performance.

##  Appendix A Proof of Lemma 1

By taking the natural logarithm on both sides of (13) and setting the lower
bound of $\text{BLEU}_{k}$, we have:

$\ln(\text{BLEU}_{k})=1-\frac{1}{\rho_{k}}+\sum_{g=1}^{G}w_{g,k}\log
p_{g,k}\geq\ln Q,$ (A.1)

with some simple algebra operations, we have:

$\rho_{k}\geq\frac{1}{1-\ln Q+\sum_{g=1}^{G}w_{g,k}\log p_{g,k}}.$ (A.2)

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# On measures induced by forcing names for ultrafilters

Piotr Borodulin–Nadzieja  Instytut Matematyczny, Uniwersytet Wrocławski, pl.
Grunwaldzki 2/4, 50-384 Wrocław, Poland<EMAIL_ADDRESS>and
Katarzyna Cegiełka  Katedra Matematyki i Cybernetyki, Uniwersytet Ekonomiczny
we Wrocławiu, ul. Komandorska 118/120, 53-345 Wrocław, Poland
<EMAIL_ADDRESS>
###### Abstract.

We study the interplay between properties of measures on a Boolean algebra
$\mathbb{A}$ and forcing names for ultrafilters on $\mathbb{A}$. We show that
several well known measure theoretic properties of Boolean algebras (such as
supporting a strictly positive measure or carrying only separable measures)
have quite natural characterizations in the forcing language. We show some
applications of this approach. In particular, we reprove a theorem of Kunen
saying that in the classical random model there are no towers of height
$\omega_{2}$.

###### 2010 Mathematics Subject Classification:

03E05, 03E75, 46B15, 46B45

The first author was supported by National Science Center project no.
2018/29/B/ST1/00223. .

Dedicated to the memory of Kenneth Kunen (1943 – 2020).

In this article we try to harness measure theory to understand better models
obtained by adding random reals.

One of the main inconveniences in forcing _praxis_ lies in dealing with names
for objects in the extension. Formally, names are quite complicated objects
defined by recursion. Fortunately, in many situations, one can find a way to
handle this difficulty. E.g. for many forcing notions we may think about names
for reals as Borel functions $f\colon 2^{\omega}\to 2^{\omega}$ in the ground
model. In this article we exploit a way to treat names for ultrafilters. If
$\mathbb{A}$ is a Boolean algebra from the ground model and
$\dot{\mathcal{U}}$ is a $\mathbb{P}$-name for an ultrafilter on $\mathbb{A}$,
then we may think about $\dot{\mathcal{U}}$ as a Boolean homomorphisms
$\varphi\colon\mathbb{A}\to\mathbb{P}$ (see the beginning of Section 2).

This approach turns out to be quite fruitful in case when
$\mathbb{P}=\mathbb{M}_{\kappa}$, i.e. the measure algebra of type $\kappa$
(see Section 1 for the precise definitions). In the article [BNS21] due to the
first author and Damian Sobota it was used to study sequences of ultrafilters
in the forcing extensions by measure algebras.

Here, we will rather focus on properties of single names. Every Boolean
homomorphism $\varphi\colon\mathbb{A}\to\mathbb{M}_{\kappa}$ naturally induces
a measure $\mu$ on $\mathbb{A}$, the ’pre-image’ of the standard Haar measure
on $\mathbb{M}_{\kappa}$. The question is how much information about a name
for an ultrafilter we get from the measure induced by the appropriate
homomorphism. We discuss this and related questions in Section 2. We present
several examples of names for ultrafilters on various Boolean algebras showing
how the properties of the induced measures affect the properties of the
ultrafilters in the extension. We show that under $\mathsf{GCH}$ in every
family of size of $\omega_{2}$ of homomorphisms from the Cantor algebra to
$\mathbb{M}_{\omega_{2}}$ one can ’swap’ two of them by an automorphism of
$\mathbb{M}_{\omega_{2}}$ (Theorem 2.6). In a sense this means that sets of
real of size $\mathfrak{c}$ in the classical random model have to be in a
sense homogeneous . Using this result we reprove the famous Kunen’s theorem
saying that there are no $\omega_{2}$ chains in $\mathcal{P}(\omega)/Fin$ in
the classical random model (Theorem 2.7). Also, we show that new ultrafilters
on $\mathcal{P}(\omega)/Fin\cap V$, where $V$ denotes the ground model, cannot
be $\sigma$-closed (Proposition 2.5).

Section 3 is devoted to so called Measure Recognition Problems and its
relations to random forcing. In [D0̌6] Džamonja coined a notion of Measure
Recognition Problems to describe several problems considered in measure
theory. Generally speaking, Measure Recognition Problem for a property
$\varphi$ ($\mathsf{MRP}$($\varphi$)) is a question about a combinatorial
characterization of Boolean algebras carrying measures having property
$\varphi$111In [D0̌6] Džamonja assumed additionally that the measure in
question should be strictly positive. Here, we drop this assumption.. We will
briefly overview the most important examples.

  * •

strict positivity. In [Kel59] Kelley fully characterized Boolean algebras
supporting strictly positive measures in a combinatorial way (using so called
Kelly intersection numbers).

  * •

$\sigma$-additivity & strict positivity. The question how to characterize
Boolean algebras with strictly positive $\sigma$-additive measures has a long
history and it was a driving force for set theoretic measure theory for many
years. It started with the works of von Neumann and Maharam ([Mah47]) and
ended in 2006 with an article by Talagrand ([Tal08]) with a solution to so
called Control Measure Problem. See [D0̌6] for the more detailed history.

  * •

separability & strict positivity. The question when a Boolean algebra
$\mathbb{A}$ supports a separable measure was considered by Mägerl and Namioka
([MN80]) and Talagrand ([Tal80]). It is known that this property is very close
to so called approximability, a purely combinatorial notion which itself is
equivalent to weak∗ separability of the space of measures on $\mathbb{A}$.
However, there is a $\mathsf{ZFC}$ example of an approximable Boolean algebra
which does not support a separable measure (see [DP08]).

  * •

non-separability. In [DK95] Džamonja and Kunen studied compact spaces carrying
only separable measures calling it _measure separable_ spaces. We will adapt
this terminology for Boolean algebras. Note that if a Boolean algebra contains
an uncountable independent family, then it is not measure separable. There are
many consistent examples (see e.g. [Kun81]) of Boolean algebras without
uncountable independent families which do carry a nonseparable measure. The
question if consistently every Boolean algebra without uncountable independent
family is measure separable was known as the Haydon Problem. It was answered
in positive by Fremlin ([Fre97]): such a situation holds under
$\mathsf{MA}_{\omega_{1}}$.

  * •

non-atomicity. By theorem of Pełczyński and Semadeni a Boolean algebra carries
an atomless measure if and only if it is not superatomic (equivalently, if its
Stone space is not scattered), see [PS59].

In Section 3 we obtain characterizations of some of the properties mentioned
above in forcing terms. We characterize Boolean algebras with strictly
positive measure by the property of being a $\sigma$-centered Boolean algebra
in some random extension (Theorem 3.2), reproving a result of Kamburelis from
[Kam89]. Using the homomorphism form of names we were able to make use of a
result from ergodic theory and to simplify the original proof of Kamburelis.
Also, we deal with $\mathsf{MRP}$(non-atomicity) showing that Boolean algebras
carrying an atomless measures are exactly those which do not have new
ultrafilters in the random extension (Proposition 3.5). We show that a Boolean
algebra carries a non-separable measure if and only if there is a
$\mathbb{M}_{\omega_{1}}$-name for a ultrafilter which cannot be added by the
forcing adding a single random real (Theorem 3.9). We recall certain Kunen’s
construction to show that consistently there is a ’small’ Boolean algebra
$\mathbb{A}$ such that forcing with a single random real cannot add ’too
generic’ ultrafilters on $\mathbb{A}$ (Corollary 3.12).

In Section 4 we focus on sequences of homomorphisms into measure algebras
ignoring its forcing interpretation. We show that on every infinite Boolean
algebra one can define a sequence of homomorphisms into the measure algebra
which is in a sense non-trivially convergent: it converges pointwise but not
uniformly (Theorem 4.6).

##  1\. Preliminaries

We consider non-zero finitely additive measures on Boolean algebras. The
following theorem says that each such measure can be uniquely extended to a
$\sigma$-additive Radon measure on the Stone space of $\mathbb{A}$.

######  Theorem 1.1.

(see e.g. [Lac74, Chapter 5, Theorem 3]) Let $\mathbb{A}$ be a Boolean algebra
and let $K$ be its Stone space. Every (finitely additive) measure $\mu$ on
$\mathbb{A}$ can be uniquely extended (as a measure on $\mathrm{Clop}(K)$) to
a Radon ($\sigma$-additive) measure $\hat{\mu}$ defined on $K$. Moreover, for
each $\varepsilon>0$ and a Borel set $B\subseteq K$, there is a clopen
$A\subseteq K$ such that $\hat{\mu}(B\triangle A)<\varepsilon$.

In what follows, for a measure $\mu$ on a Boolean algebra we will denote by
$\hat{\mu}$ the extension promised by the above theorem.

For a Boolean algebra $\mathbb{A}$ let
$\mathbb{A}^{+}=\mathbb{A}\setminus\\{0\\}$. We say that a Boolean algebra
$\mathbb{A}$ _supports a measure_ $\mu$ if $\mu(A)>0$ for each
$A\in\mathbb{A}^{+}$ (in this case we say that $\mu$ is _strictly positive_ on
$\mathbb{A}$). If $\mu$ is a measure on $\mathbb{A}$, which is not necessarily
strictly positive, then we say that $\mathbb{A}$ _carries_ $\mu$.

For a cardinal $\kappa$ denote by $\lambda_{\kappa}$ the standard (product)
measure on $\\{0,1\\}^{\kappa}$. Let
$\mathbb{M}_{\kappa}=\mathrm{Bor}(\\{0,1\\}^{\kappa})/_{\lambda_{\kappa}=0}$.
We will call $\mathbb{M}_{\kappa}$ _measure algebras_. Note that
$\mathbb{M}_{1}$ is the trivial algebra and $\mathbb{M}_{\omega}$ is usually
called _the measure algebra_ (and will be denoted by $\mathbb{M}$). By saying
that we add $\kappa$ ($\kappa>\omega$) random reals, we mean that we force
with $\mathbb{M}_{\kappa}$. Adding a single random real means forcing with
$\mathbb{M}$.

A measure $\mu$ on $\mathbb{A}$ is _atomless_ if its extension to $\hat{\mu}$
vanishes on points. Equivalently, $\mu$ is atomless if for each
$\varepsilon>0$ there is a partition of unity $(A_{n})_{n<N}$ such that
$\mu(A_{n})<\varepsilon$ for every $n<N$.

If $\mathbb{A}$ carries a measure $\mu$, then the function
$d_{\mu}\colon\mathbb{A}\to\mathbb{R}$ defined by $d_{\mu}(A,B)=\mu(A\triangle
B)$ is a pseudo-metric (which is a metric, if $\mu$ is strictly positive). We
will call it _a Frechet-Nikodym (pseudo-)metric_. By the _Maharam type of
$\mu$_ we understand the density of the pseudometric space
$(\mathbb{A},d_{\mu})$. We say that a measure $\mu$ on a Boolean algebra
$\mathbb{A}$ is _homogeneuous_ if $\mu_{|A}$ has the same Maharam type for
each $A\in\mathbb{A}^{+}$.

We say that a homomorphism $\varphi\colon(\mathbb{A},\mu)\to(\mathbb{B},\nu)$
is _metric_ if $\nu(\varphi(A))=\mu(A)$ for each $A\in\mathbb{A}$. We will say
that a Boolean algebra $\mathbb{A}$ is metrically isomorphic to $\mathbb{B}$
if it is clear which measures on $\mathbb{A}$ and $\mathbb{B}$ we mean.

######  Theorem 1.2 (Maharam’s theorem).

(see [Mah42]) If a Boolean algebra $\mathbb{A}$ supports a countably additive
measure $\mu$, then $(\mathbb{A},\mu)$ is metrically isomorphic to a direct
sum of measure algebras (with its standard measures). If $\mu$ is homogeneous
of Maharam type $\kappa$, then $(\mathbb{A},\mu)$ is metrically isomorphic to
$\mathbb{M}_{\kappa}$.

A measure $\mu$ on a Boolean algebra is _separable_ if $(\mathbb{A},d_{\mu})$
is separable. Note that $\mu$ on $\mathbb{A}$ is separable if and only if the
space $L_{1}(\hat{\mu})$ is separable. It will be convenient to make the
following consequence of Maharam’s theorem available.

######  Proposition 1.3.

Assume that $\mu$ is a stricly positive separable measure on $\mathbb{A}$.
Then there is an injective metric homomorphism
$\varphi\colon(\mathbb{A},\mu)\to(\mathbb{M},\lambda)$.

###### Proof.

Let $K$ be the Stone space of $\mathbb{A}$. Notice that if $\mu$ is separable,
then $\hat{\mu}$ is separable on $\mathrm{Bor}(K)_{/\hat{\mu}=0}$ (since Borel
sets can be approximated with respect to $\hat{\mu}$ with clopens). Then, by
Theorem 1.2, there is a metric isomorphism
$\varphi^{\prime}\colon\mathrm{Bor}(K)_{/\hat{\mu}=0}\to\mathbb{M}$. Let
$\varphi=\varphi^{\prime}_{|\mathrm{Clop}(K)}$. ∎

A Boolean algebra $\mathbb{A}$ is _$\sigma$ -centered_ if $\mathbb{A}^{+}$ is
a union of countably many ultrafilters on $\mathbb{A}$, or, equivalently, if
$\mathrm{St}(\mathbb{A})$ is separable. Note that if $\mathbb{A}$ is
$\sigma$-centered Boolean algebra witnessed by $\\{\mathcal{U}_{n}\colon
n\in\omega\\}$, then $\mu=\sum\delta_{\mathcal{U}_{n}}/2^{n+1}$ is strictly
positive on $\mathbb{A}$ (and so, $\mathbb{A}$ supports a measure).

In general we denote elements of Boolean algebras, including
$\mathbb{M}_{\kappa}$’s, with capital letters. However, speaking about an
element of $\mathbb{M}_{\kappa}$ sometimes we understand it as a condition of
the forcing notion $\mathbb{M}^{+}_{\kappa}$. In this case, we will rather use
the standard forcing notation: $p$, $q$ and so on.

By _a real_ we will understand an element of the Cantor set $2^{\omega}$ (or,
equivalently, by the standard identification of $2^{\omega}$ with
$\mathcal{P}(\omega)$, a subset of $\omega$).

For unexplained terminology consult [Kun83] and [BJ95] (forcing), [Hal50]
(measure theory).

##  2\. Homomorphisms and names for ultrafilters

Suppose that we force by $\mathbb{M}_{\kappa}$ over a model $V$. Let
$\mathbb{A}$ be a Boolean algebra in $V$. Then, in $V[G]$, $\mathbb{A}$ also
forms a Boolean algebra. (Formally, we should speak here about the canonical
name $\check{\mathbb{A}}$ for $\mathbb{A}$ but we are going to abuse the
notation by identifying objects from the ground model with its canonical
names.) Typically, in $V[G]$ there are new ultrafilters on $\mathbb{A}$.

A priori the access to those ’new’ ultrafilters is quite remote. However, the
point is that there is a natural correspondence between
$\mathbb{M}_{\kappa}$-names of ultrafilters on $\mathbb{A}$ and homomorphisms
$\varphi\colon\mathbb{A}\to\mathbb{M}_{\kappa}$. Indeed, assume
$\dot{\mathcal{U}}$ is a $\mathbb{M}_{\kappa}$-name for an ultrafilter on a
Boolean algebra $\mathbb{A}$. Define
$\varphi\colon\mathbb{A}\to\mathbb{M}_{\kappa}$ by $\varphi(A)=\llbracket
A\in\dot{\mathcal{U}}\rrbracket$, where $\llbracket\alpha\rrbracket$ is the
truth value of $\alpha$ (see e.g. [Kun81, Definition 7.14]). It is plain to
check that it is a Boolean homomorphism. On the other hand, fix a homomorphism
$\varphi\colon\mathbb{A}\to\mathbb{M}_{\kappa}$ and let

$\dot{\mathcal{U}}=\\{\langle A,\varphi(A)\rangle\colon A\in\mathbb{A}\\}.$

Then $\dot{\mathcal{U}}$ is an $\mathbb{M}_{\kappa}$-name for an ultrafilter
on $\mathbb{A}$.

So, every homomorphism $\varphi\colon\mathbb{A}\to\mathbb{M}_{\kappa}$ can be
interpreted as an $\mathbb{M}_{\kappa}$-name for an ultrafilter on
$\mathbb{A}$ (and will be denoted by $\dot{\varphi}$ in this case). Also, if
$\dot{\mathcal{U}}$ is an $\mathbb{M}_{\kappa}$-name for an ultrafilter on
$\mathbb{A}$, then we may assume that it is of the form $\dot{\varphi}$ for
some homomorphism $\varphi$ (since
$\Vdash_{\mathbb{M}_{\kappa}}\dot{\varphi}=\dot{\mathcal{U}}$, where $\varphi$
is defined by $\varphi(A)=\llbracket A\in\dot{U}\rrbracket$).

If $\varphi\colon\mathbb{A}\to\mathbb{M}_{\kappa}$ is a Boolean homomorphism,
then $\mu=\lambda\circ\varphi$ is a measure. We will call it _the measure
induced_ by $\varphi$. In this way, each $\mathbb{M}_{\kappa}$-name
$\dot{\varphi}$ for ultrafilter on $\mathbb{A}$ induces a measure on
$\mathbb{A}$.

######  Example 2.1.

If the range of a homomorphism $\varphi\colon\mathbb{A}\to\mathbb{M}$ is
trivial, then $\mathcal{U}=\varphi^{-1}[\\{1\\}]$ is an ultrafilter on
$\mathbb{A}$ (in $V$). In this case $\dot{\varphi}$ is a canonical name for
$\mathcal{U}$ and $\lambda\circ\varphi=\delta_{\mathcal{U}}$, the Dirac delta
at $\mathcal{U}$. See also Proposition 3.4.

The next simple example shows that there may be two names for ultrafilters
which induce the same measure but which are interpreted as distinct points by
each generic filter.

######  Example 2.2.

Consider a Boolean algebra $\mathbb{A}$ and
$\mathcal{U}_{0}\neq\mathcal{U}_{1}\in\mathrm{St}(\mathbb{A})$. Let
$M\in\mathbb{M}$ be such that $\lambda(M)=1/2$ and for $i\in\\{0,1\\}$ define
$\varphi_{i}\colon\mathbb{A}\to\mathbb{M}$

$\varphi_{i}(A)=\begin{cases}1\mbox{ if
}A\in\mathcal{U}_{1}\cap\mathcal{U}_{0},\\\ M\mbox{ if
}A\in\mathcal{U}_{i}\setminus\mathcal{U}_{1-i},\\\ M^{c}\mbox{ if
}A\in\mathcal{U}_{1-i}\setminus\mathcal{U}_{i},\\\ 0\mbox{ otherwise.
}\end{cases}$

Then $\varphi_{0}$ and $\varphi_{1}$ induce the same measure:

$\lambda\circ\varphi_{0}=\lambda\circ\varphi_{1}=1/2(\delta_{\mathcal{U}_{0}}+\delta_{\mathcal{U}_{1}}),$

but $\Vdash_{\mathbb{M}}\dot{\varphi_{0}}\neq\dot{\varphi_{1}}$.

The above example shows that we have to be careful if we want to deduce
something about the relationship of names for ultrafilters by looking at the
induced measures.

Perhaps the most important Boolean algebra to study in this context is the
Cantor algebra, as its ultrafilters can be interpreted as reals.

######  Example 2.3.

Consider the Cantor algebra $\mathbb{C}=\mathrm{Clop}(2^{\omega})$. Let
$\varphi\colon\mathbb{C}\to\mathbb{M}$ be a homomorphism sending clopens to
its equivalence classes in $\mathbb{M}$. Then $\dot{\varphi}$ is a name for
the generic random real (see [Kun84, Definition 3.9]). In general, suppose
that $\varphi\colon\mathbb{C}\to\mathbb{M}_{\kappa}$ is a homomorphism and
$\mu=\lambda_{\kappa}\circ\varphi$. If $\mu$ is atomless, then $\dot{\varphi}$
is a name for a ’new’ real in the following sense: for each $x\in 2^{\omega}$
there is $p\in\mathbb{M}_{\kappa}$ such that $p\Vdash x\neq\dot{\varphi}$.
Indeed, take $C\in\mathbb{C}$ such that $x\notin C$ and $\mu(C)>0$. Then
$\varphi(C)\Vdash x\neq\dot{\varphi}$.

######  Remark 2.4.

Recall that if $\dot{r}$ is a random real then it avoids measure zero sets
coded in the ground model (see [Kun84, Lemma 3.8]). An example of application
of this fact is the following. If ($\alpha$) is a Borel property of real
numbers such that $\\{x\in 2^{\omega}\colon x$ has ($\alpha$)$\\}$ is of full
measure, then $\Vdash_{\mathbb{M}_{\kappa}}\dot{r}$ possess ($\alpha$), where
$\dot{r}$ is the generic random real. So, e.g., Strong Law of Large Numbers
implies that $\Vdash_{\mathbb{M}_{\kappa}}$ $\dot{r}$ has asymptotic density
1/2 (seen as a subset of $\omega$). Using measures we can find names for new
reals which have some additional properties. Let $(\alpha)$ be a Borel
property of reals such that $F=\\{x\in 2^{\omega}\colon x\mbox{ has
}(\alpha)\\}$ is not scattered. Then there is an atomless measure $\mu$ on
$\mathbb{C}$ such that $\hat{\mu}(F)=1$. By Proposition 1.3, there is $\kappa$
and a homomorphism $\varphi\colon\mathbb{C}\to\mathbb{M}_{\kappa}$ such that
$\mu=\lambda_{\kappa}\circ\varphi$. Then $\dot{\varphi}$ is a name for a new
real (see Example 2.3) such that
$\Vdash_{\mathbb{M}_{\kappa}}\dot{\varphi}\mbox{ has }(\alpha)$.

The next example will concern the algebra $\mathcal{P}(\omega)$. The Boolean
algebra of those subsets of $\omega$ which belongs to the ground model is a
subalgebra of $\mathcal{P}(\omega)$ in the forcing extension (which is a
proper subalgebra if the forcing adds new reals). The Stone spaces of such
Boolean algebras of ’old’ subsets of $\omega$ were considered e.g. in [SZ19]
and in the context of random forcing in [DF07]. The measures on
$\mathcal{P}(\omega)$ were studied e.g. in [FT79]. Note that there are many
atomless measures on $\mathcal{P}(\omega)$, although none of them can be
strictly positive. A typical example of such a measure is of the form
$\mu(A)=\lim_{n\to\mathcal{U}}|A\cap n|/n$, where $\mathcal{U}$ is an
ultrafilter on $\mathcal{P}(\omega)$ (see [BFPRN01]). Such measure extends the
asymptotic density.

Recall that an ultrafilter $\mathcal{U}$ on $\mathcal{P}(\omega)$ is a P-point
if $\mathcal{U}$ is $\sigma$-closed with respect to $\subseteq^{*}$, i.e.
inclusion modulo finite sets. Notice that this definition makes sense also for
subalgebras of $\mathcal{P}(\omega)$, in particular of
$\mathcal{P}(\omega)\cap V$ if we work in a forcing extension. The following
fact says that the new ultrafilters on $\mathcal{P}(\omega)\cap V$ cannot be
P-points.

######  Proposition 2.5.

Suppose that $\varphi\colon\mathcal{P}(\omega)\to\mathbb{M}$ is such that
$\mu=\lambda\circ\varphi$ is atomless. Then $\Vdash_{\mathbb{M}}\dot{\varphi}$
is not a P-point on $\mathcal{P}(\omega)\cap V$.

###### Proof.

Denote $\mathbb{A}=\mathcal{P}(\omega)$. Since $\mu$ is atomless for every $n$
we can find a maximal finite antichain $\mathcal{A}_{n}$ in $\mathbb{A}$ such
that $\mu(A)<1/n$ for each $A\in\mathcal{A}_{n}$. Let $G$ be a generic filter
on $\mathbb{M}$. From this point on we will work in $V[G]$, understanding
$\mathbb{A}$ as $\mathcal{P}(\omega)\cap V$. Notice that for each $n$ there is
exactly one $A_{n}\in\mathcal{A}_{n}$ such that $\varphi(A_{n})\in G$. Then
$(A_{n})$ is a sequence of elements of $\dot{\varphi}$. But if
$A\in\mathbb{A}$ is such that $A\subseteq^{*}A_{n}$ for each $n$, then
$\mu(A)=0$ (because $\mu$ is atomless). Hence $\varphi(A)=0$ and so $1\Vdash
A\notin\dot{\varphi}$. Thus, $(A_{n})$ witnesses that $\dot{\varphi}$ is not a
P-point. ∎

Note that Kunen in [Kun76] used measures on $\omega$ in a slightly different
way to show that in the random model there are no selective ultrafilters.

Now we will prove a result showing that in the forcing extension by
$\mathbb{M}_{\omega_{2}}$ over a model satisfying $\mathsf{GCH}$ (so in the
classical random model) the sets of reals of size $\mathfrak{c}$ have to be in
a sense homogeneous.

######  Theorem 2.6.

Assume $GCH$. Let $(\varphi_{\alpha})_{\alpha<\omega_{2}}$ be a family of
homomorphisms $\varphi_{\alpha}\colon\mathbb{C}\to\mathbb{M}_{\omega_{2}}$.
Then there are $\alpha<\beta<\omega_{2}$ and an automorphism
$\Phi\colon\mathbb{M}_{\omega_{2}}\to\mathbb{M}_{\omega_{2}}$ such that
$\Phi\circ\varphi_{\alpha}(C)=\varphi_{\beta}(C)$ and
$\Phi\circ\varphi_{\beta}(C)=\varphi_{\alpha}(C)$ for every $C\in\mathbb{C}$.

###### Proof.

First, fix an independent family $\\{C_{n}\colon
n\in\omega\\}\subseteq\mathbb{C}$ generating $\mathbb{C}$. We say that
$A\in\mathbb{C}$ is _a chunk_ if it is of the form

$A=(C_{i_{0}}\land\dots\land C_{i_{k}})\land(C^{c}_{j_{0}}\land\dots\land
C^{c}_{j_{l}})$

for sequences $(i_{n})_{n\leq k}$, $(j_{n})_{n\leq l}$ of integers.

We say that $\alpha$ and $\beta$ are _symmetric_ if

$\varphi_{\alpha}(A)\land\varphi_{\beta}(B)=0\iff\varphi_{\beta}(A)\land\varphi_{\alpha}(B)=0$

for each chunks $A$ and $B$.

Claim. There are distinct $\alpha$, $\beta<\omega_{2}$ which are symmetric.

Suppose the opposite. Then for each $\alpha<\beta$ we can choose chunks $A$,
$B$ and $i\in\\{0,1\\}$ such that

$i=0\mbox{ and }\varphi_{\alpha}(A)\land\varphi_{\beta}(B)=0\mbox{ and
}\varphi_{\beta}(A)\land\varphi_{\alpha}(B)\neq 0$

or

$i=1\mbox{ and }\varphi_{\alpha}(A)\land\varphi_{\beta}(B)\neq 0\mbox{ and
}\varphi_{\beta}(A)\land\varphi_{\alpha}(B)=0.$

Since $\mathbb{C}\times\mathbb{C}\times\\{0,1\\}$ is countable, by GCH and
Erdős-Rado theorem
$\omega_{2}=2^{2^{\omega}}\rightarrow(\omega^{+})^{2}_{\omega}$, we could find
an uncountable set $\Lambda\subseteq\omega_{2}$ such that for each
$\alpha,\beta\in\Lambda$ we have assigned the same configuration of chunks
$A$, $B$ and $i\in\\{0,1\\}$. We may assume without loss of generality that
$\Lambda=\omega_{1}$, $i=0$ and so for each $\alpha<\beta<\omega_{1}$ we have

$\varphi_{\alpha}(A)\land\varphi_{\beta}(B)=0\mbox{ and
}\varphi_{\beta}(A)\land\varphi_{\alpha}(B)\neq 0.$

Let $D=\bigvee_{\alpha<\omega_{1}}\varphi_{\alpha}(A)$. There is
$\gamma<\omega_{1}$ such that $\bigvee_{\alpha<\gamma}\varphi_{\alpha}(A)=D$.
Then $D\land\varphi_{\gamma}(B)=0$. Let $\beta>\gamma$. Then
$\varphi_{\beta}(A)\land\varphi_{\gamma}(B)\neq 0$ but $\varphi_{\beta}(A)\leq
D$, a contradiction.

So, let $\alpha<\beta<\omega_{2}$ be symmetric. Define $\Phi$ on
$\mathcal{A}=\\{\varphi_{i}(C_{n})\colon n\in\omega,i\in\\{\alpha,\beta\\}\\}$
by setting $\Phi(\varphi_{\alpha}(C_{n}))=\varphi_{\beta}(C_{n})$ and
$\Phi(\varphi_{\beta}(C_{n}))=\varphi_{\alpha}(C_{n})$. Then $\Phi$ can be
extended to an automorphism of the Boolean algebra generated by $\mathcal{A}$.
Indeed, by Sikorski’s Extension Criterion (see [KMB89, Proposition 5.6]) it is
enough to check that

$\varphi_{\alpha}(A)\land\varphi_{\beta}(B)=0\iff(\Phi(\varphi_{\alpha}(A))\land\Phi(\varphi_{\beta}(B))=0.$

But since $\Phi\circ\varphi_{\alpha}=\varphi_{\beta}$ and
$\Phi\circ\varphi_{\beta}=\varphi_{\alpha}$ the above condition translates to
symmetry of $\alpha$ and $\beta$.

Then $\Phi$ can be extended to an automorphism of the whole
$\mathbb{M}_{\omega_{2}}$ (see [Fre04, proof of (a) in Theorem 383E], [Fre04,
Theorem 333C(b)]). ∎

As a corollary we will reprove one of the famous results of Kunen. We will
show that in the random model there are no towers of size $\omega_{2}$. In
fact, the Kunen’s original theorem is about the Cohen model. Kunen’s proof
seems to belong to the oral part of the set theoretic tradition and we are not
aware of any literature containing it (apart perhaps of [Kun68]).

######  Theorem 2.7 (Kunen).

Let $V$ be the model of $GCH$ and let $G$ be the
$\mathbb{M}_{\omega_{2}}$-generic. Then, in $V[G]$, there is no well-ordered
chain of size $\omega_{2}$ in $\mathcal{P}(\omega)/Fin$.

###### Proof.

Suppose that there is a strictly $\subseteq^{*}$-increasing chain of size
$\omega_{2}$ in $V[G]$. Then there is a sequence
$(\varphi_{\alpha})_{\alpha<\omega_{2}}$ of homomorphisms
$\varphi_{\alpha}\colon\mathbb{C}\to\mathbb{M}_{\omega_{2}}$ such that

$\Vdash_{\mathbb{M}_{\omega_{2}}}(\dot{\varphi}_{\alpha})_{\alpha<\omega_{2}}\mbox{
is a }\subseteq^{*}\mbox{-chain of subsets of }\omega.$

We will work in $V$. By Theorem 2.6 there are $\alpha<\beta<\omega_{2}$ and an
automorphism $\Phi\colon\mathbb{M}_{\omega_{2}}\to\mathbb{M}_{\omega_{2}}$
such that $\Phi\circ\varphi_{\alpha}=\varphi_{\beta}$ and
$\Phi\circ\varphi_{\beta}=\varphi_{\alpha}$. Then

$\Phi[\mathbb{M}_{\omega_{2}}]\Vdash\Phi\circ\dot{\varphi}_{\alpha}\subseteq^{*}\Phi\circ\dot{\varphi}_{\beta},$

and so

$\mathbb{M}_{\omega_{2}}=\Phi[\mathbb{M}_{\omega_{2}}]\Vdash\dot{\varphi}_{\beta}\subseteq^{*}\dot{\varphi}_{\alpha},$

which clearly leads to the contradiction with the assumption that the chain is
strictly increasing. ∎

######  Remark 2.8.

Every $\mathbb{M}$-name for an ultrafilter on $\mathbb{M}$ itself can be seen
as a homomorphism $\varphi\colon\mathbb{M}\to\mathbb{M}$. The family of all
such homomorphisms with the composition is a semi-group. So, in $V[G]$, where
$G$ is an $\mathbb{M}$-generic, we have a semi-group structure on the Stone
space of $\mathbb{M}\cap V$. It seems that this structure has no natural
counterpart in the Stone space of $\mathbb{M}$ in the ground model. In the
next section we will see that this structure with a little bit of ergodic
theory may help us to prove some theorems about the Stone space of
$\mathbb{M}\cap V$.

##  3\. Measure Recognition Problems in forcing context

In this section we overview some of the properties used in the Measure
Recognition Problems mentioned in the introduction. We will show that those
properties have characterization in terms of forcing with measure algebras.

###  3.1. Boolean algebras supporting measures

The question about a characterization of Boolean algebras supporting measures
has a long history. In case we ask this question in the context of
$\sigma$-additive measures, it leads to the famous Maharam’s Problem. The
combinatorial charaterization for finitely additive measures is much less
mysterious; it was formulated by Kelley in [Kel59].

In [Kam89] Kamburelis proved an elegant characterization of Boolean algebras
supporting measures in the forcing terminology (Theorem 3.2 below). If we look
at the names for ultrafilters as homomorphisms, the proof almost trivializes
provided we use a well-known fact from the theory of dynamical systems.

Let $(X,\Sigma,\mu)$ be a measure space. We say that $T\colon X\to X$ is a
_measure-preserving transformation_ if $\mu(T[A])=\mu(A)$ for each measurable
$A$.

A measure-preserving transformation $T$ is called _ergodic_ if whenever $A$ is
invariant, i.e. $T^{-1}[A]=A$, then $\mu(A)=0$ or $\mu(A)=1$. Note that this
is equivalent to saying that whenever $A,B$ are $\mu$-positive then
$\mu(T^{-n}[A]\cap B)>0$ for some $n$. It follows from the fact that
$\bigcup_{n}T^{-n}[A]$ is invariant and so $\mu(\bigcup_{n}T^{-n}[A])=1$.

An example of an ergodic transformation of
$([0,1],\mathrm{Bor}([0,1]),\lambda)$ is $T(x)=x+r$, where $r$ is irrational
and $+$ is taken modulo $1$. We will use the fact that for each infinite
$\kappa$ there is an ergodic transformation of
$([0,1]^{\kappa},\mathrm{Bor}([0,1]^{\kappa}),\lambda_{\kappa})$.

######  Proposition 3.1.

[CP82, Section 1.4] Let $\kappa$ be an infinite cardinal number. There is a
metric automorphism $\varphi\colon\mathbb{M}_{\kappa}\to\mathbb{M}_{\kappa}$
such that whenever $A,B\in\mathbb{M}^{+}_{\kappa}$, there is $n\in\omega$ such
that $\varphi^{n}(A)\cap B\neq 0$.

###### Proof.

Let $X=[0,1]^{\kappa}$. Consider $X^{\mathbb{Z}}$ with the product measure
$\mu$. Let $T\colon X^{\mathbb{Z}}\to X^{\mathbb{Z}}$ be the Bernoulli shift,
i.e. $T(x)(n)=x(n+1)$. Then, for every measurable $A,B\subseteq
X^{\mathbb{Z}}$ we have $\lim_{n\to\infty}\mu(T^{-n}[A]\cap
B)=\mu(A)\cdot\mu(B)>0$ (i.e. Bernoulli shift is mixing, see e.g. [Shi73,
Chapter 3]) and so $T$ is ergodic.

Define a metric automorphism $\varphi$ of
$\mathrm{Bor}(X^{\mathbb{Z}})/_{\mu=0}$ by $\varphi(A)=T^{-1}[A]$ and notice
that, by Maharam’s theorem, $(\mathrm{Bor}(X^{\mathbb{Z}})/_{\mu=0},\mu)$ is,
in fact, isomorphic to $(\mathbb{M}_{\kappa},\lambda_{\kappa})$ and so we may
think that $\varphi$ is an automorphism of $\mathbb{M}_{\kappa}$. Clearly,
$\varphi$ satisfies the desired property. ∎

######  Theorem 3.2.

[Kam89, Proposition 3.7] Let $\mathbb{A}$ be a Boolean algebra. Then the
following are equivalent

  * •

$\mathbb{A}$ supports a (strictly positive) measure,

  * •

there is $\kappa$ such that $1\Vdash_{\mathbb{M}_{\kappa}}\mathbb{A}$ is
$\sigma$-centered.

###### Proof.

Assume that $\mathbb{A}$ supports a measure $\mu$. Then, by Proposition 1.3,
there is a metric homomorphism $\psi\colon\mathbb{A}\to\mathbb{M}_{\kappa}$
for some $\kappa$. Let
$\varphi\colon\mathbb{M}_{\kappa}\to\mathbb{M}_{\kappa}$ be as in Proposition
3.1. Denote $\rho_{n}=\varphi^{n}\circ\psi$. We claim that

$1\Vdash_{\mathbb{M}_{\kappa}}\mathbb{A}^{+}=\bigcup_{n}\dot{\rho}_{n}.$

Indeed, let $A\in\mathbb{A}^{+}$ and let $p\in\mathbb{M}^{+}_{\kappa}$. There
is $n$ such that $q=(\varphi^{n}(\psi(A))\cap p)\neq 0$. But then $q\Vdash
A\in\dot{\rho_{n}}$. By the density argument we are done.

Now suppose that $1\Vdash_{\mathbb{M}_{\kappa}}\mathbb{A}$ is
$\sigma$-centered. Then there is a family $\\{\dot{\rho}_{n}\colon
n\in\omega\\}$ of $\mathbb{M}_{\kappa}$-names for ultrafilters on $\mathbb{A}$
such that

$1\Vdash_{\mathbb{M}_{\kappa}}\mathbb{A}^{+}=\bigcup_{n}\dot{\rho}_{n}.$

For every $n$ let $\mu_{n}=\lambda\circ\rho_{n}$. Then
$\mu=\sum_{n}\mu_{n}/2^{n}$ is strictly positive on $\mathbb{A}$. Otherwise,
there is $A\in\mathbb{A}^{+}$ such that $\mu_{n}(A)=0$ for each $n$. Hence,
$\rho_{n}(A)=0$ for every $n$ and so for each $n$ we would have
$1\Vdash_{\mathbb{M}_{\kappa}}A\notin\dot{\rho}_{n}$, a contradiction. ∎

As a corollary we get a simple proof of the folklore fact saying that the
measure algebra $\mathbb{M}$ becomes $\sigma$-centered in the forcing
extension by $\mathbb{M}$. This is formulated e.g. in [BJ95, Proposition
3.2.11] without a (direct) proof. We show that in fact a dense subset of the
Stone space of the ’old’ $\mathbb{M}$ in the forcing extension may be induced
by a single object.

######  Proposition 3.3.

$\Vdash_{\mathbb{M}}\mathbb{M}\mbox{ is }\sigma\mbox{-centered}$.

###### Proof.

The proof is analogous to the first part of the proof of Theorem 3.2. In this
case we do not need Maharam’s theorem; $\psi$ is just the identity. ∎

###  3.2. Boolean algebras carrying atomless measures

The following fact is a generalization of Example 2.1.

######  Proposition 3.4.

Let $\varphi\colon\mathbb{A}\to\mathbb{M}_{\kappa}$. Assume that the measure
$\mu=\lambda_{\kappa}\circ\varphi$ is purely atomic, i.e.
$\mu=\sum_{n}a_{n}\delta_{\mathcal{V}_{n}}$ for some sequence $(a_{n})$ of
positive real numbers such that $\sum_{n}a_{n}=1$, and a sequence
$(\mathcal{V}_{n})$ of ultrafilters on $\mathbb{A}$. Then,
$1\Vdash_{\mathbb{M}_{\kappa}}\exists n\ \dot{\varphi}=\mathcal{V}_{n}$.

###### Proof.

For $n\in\omega$ define $p_{n}=\bigwedge\\{\varphi(V)\colon
V\in\mathcal{V}_{n}\\}$. It is plain to check that
$p_{n}\Vdash\dot{\varphi}=\mathcal{V}_{n}$. Notice also that $(p_{n})$ is an
antichain. If $n\neq m$ then there is
$V\in\mathcal{V}_{n}\setminus\mathcal{V}_{m}$ and so $p_{n}\leq\varphi(V)$ and
$p_{m}\leq\varphi(V)^{c}$.

To finish the proof we will show that $(p_{n})$ is a maximal antichain (then
the conclusion will follow from the standard density argument). First, since
$\mathbb{M}_{\kappa}$ is ccc and each $\mathcal{V}_{n}$ is closed under
intersections, for every $n$ there is a decreasing sequence $(V_{k})$ in
$\mathcal{V}_{n}$ such that $p_{n}=\bigwedge_{k}\varphi(V_{k})$. So,
$\lambda_{\kappa}(p_{n})\geq a_{n}$, by the continuity of $\lambda_{\kappa}$.
Thus, $\sum_{n}\lambda_{\kappa}(p_{n})\geq\sum_{n}a_{n}=1$ and so $(p_{n})$ is
maximal. ∎

In particular, if the measure induced by a name $\dot{\varphi}$ is purely
atomic, then $1\Vdash_{\mathbb{M}_{\kappa}}\dot{\varphi}$ is in the ground
model. At this point, we may formulate an easy corollary. Recall that a
Boolean algebra is _superatomic_ if each of its subalgebras contains an atom,
equivalently, if its Stone space is scattered, see Introduction.

######  Corollary 3.5.

Let $\mathbb{A}$ be a Boolean algebra. Then $\mathbb{A}$ is superatomic if and
only if
$1\Vdash_{\mathbb{M}_{\kappa}}\dot{\mathrm{St}}(\mathbb{A})=\mathrm{St}(\mathbb{A})$
for every $\kappa$.

###### Proof.

Use the fact that a Boolean algebra is superatomic if and only if all the
measures it carries are purely atomic (see [PS59]) and Proposition 3.4. ∎

######  Remark 3.6.

We get the above corollary ’for free’ from Proposition 3.4. In fact, it can be
generalized to other forcing notions. Recall also Mazurkiewicz-Sierpiński
theorem which says that compact scattered spaces are classifiable up to
homeomorphism by $\omega_{1}\times\omega$ (see [MS20] and [Day67]) and in this
sense they are absolute objects.

###  3.3. Boolean algebras carrying only separable measures

It is well known that every real added by forcing with $\mathbb{M}_{\kappa}$
can be added in fact added by forcing with $\mathbb{M}$. We will see that we
may characterize those names for ultrafilters which can be added by forcing
with $\mathbb{M}$.

######  Theorem 3.7.

Assume $\mathbb{A}$ is a Boolean algebra and let
$\varphi\colon\mathbb{A}\to\mathbb{M}_{\kappa}$ be a homomorphism such that
$\lambda_{\kappa}\circ\varphi$ is separable. Then there is
$\mathbb{M}^{\prime}\subseteq\mathbb{M}_{\kappa}$ isomorphic to $\mathbb{M}$
such that $\varphi[\mathbb{A}]\subseteq\mathbb{M}^{\prime}$. In particular
$\dot{\varphi}$ is an $\mathbb{M}^{\prime}$-name and so it can be added by
forcing with $\mathbb{M}$.

###### Proof.

Assume $\mathbb{A}$ and $\varphi$ are as above. For simplicity we will assume
that $1\Vdash_{\mathbb{M}_{\kappa}}\dot{\varphi}\notin V$ and so that $\mu$ is
atomless.

First, notice that $(\mathbb{A}/_{\mu=0},\mu)$ and
$(\varphi[\mathbb{A}],\lambda_{\kappa})$ are metrically isomorphic. Indeed,
$\varphi\colon\mathbb{A}\to\varphi[\mathbb{A}]$ is a metric epimorphism, whose
kernel coincides with the family of elements of $\mathbb{A}$ which are
$\mu$-null. Hence, by the assumption on $\mu$, the measure $\lambda_{\kappa}$
is separable on $\varphi[\mathbb{A}]$ and so
$(\varphi[\mathbb{A}],d_{\lambda_{\kappa}})$ is a separable metric space. Let
$\mathbb{M}^{\prime}$ be the closure of $\varphi[\mathbb{A}]$ in
$(\mathbb{M}_{\kappa},d_{\lambda_{\kappa}})$. Then $\mathbb{M}^{\prime}$ is a
Boolean subalgebra of $\mathbb{M}_{\kappa}$ and
$(\mathbb{M}^{\prime},d_{\lambda_{\kappa}})$ is still separable. But
$\mathbb{M}^{\prime}$ is complete as a closed subspace of
$\mathbb{M}_{\kappa}$ and so, by [AP82, Proposition 2], it is also complete as
a Boolean algebra. Thus, by Maharam’s theorem, it is isomorphic to
$\mathbb{M}$. ∎

######  Proposition 3.8.

A Boolean algebra $\mathbb{A}$ supports a separable measure if and only if
$\mathbb{A}$ can be embedded in $\mathbb{M}$.

###### Proof.

The forward implication is just Proposition 1.3 and the reverse implication
follows from the fact that subspaces of separable metric spaces are separable.
∎

Recall that a Boolean algebra is measure separable if it carries only
separable measure (see also Introduction).

######  Theorem 3.9.

Let $\mathbb{A}$ be a Boolean algebra. Then the following conditions are
equivalent:

  * •

$\mathbb{A}$ is measure separable,

  * •

for every $\kappa\geq\omega$ and every $\mathbb{M}_{\kappa}$-name
$\dot{\varphi}$ for an ultrafilter on $\mathbb{A}$ there is a subalgebra
$\mathbb{M}^{\prime}\subseteq\mathbb{M}_{\kappa}$ isomorphic to $\mathbb{M}$
and such that $\dot{\varphi}$ is an $\mathbb{M}^{\prime}$-name for an
ultrafilter on $\mathbb{A}$.

###### Proof.

If $\mathbb{A}$ is measure separable and
$\varphi\colon\mathbb{A}\to\mathbb{M}_{\kappa}$ is a homomorphism, then
$\lambda_{\kappa}\circ\varphi$ is separable and we can use Proposition 3.7.

Suppose now that $\mathbb{A}$ is not a measure separable Boolean algebra and
let $\mu$ be a non-separable measure on $\mathbb{A}$. Let $K$ be the Stone
space of $\mathbb{A}$. We may assume without loss of generality that
$\mathrm{Bor}(K)/_{\hat{\mu}=0}$ is metrically isomorphic to
$\mathbb{M}_{\kappa}$ for some $\kappa>\omega$. Assume that
$\varphi^{\prime}\colon\mathrm{Bor}(K)_{/\hat{\mu}=0}\to\mathbb{M}_{\kappa}$
witnesses this isomorphism. Let
$\varphi\colon\mathbb{A}\to\mathbb{M}_{\kappa}$ be defined by
$\varphi(A)=\varphi^{\prime}([A]_{\hat{\mu}=0})$. Then
$\varphi[\mathbb{A}]=\mathbb{M}_{\kappa}$ and so $\dot{\varphi}$ is not a
$\mathbb{M}^{\prime}$-name for any proper subalgebra of $\mathbb{M}_{\kappa}$.

∎

The above theorem says that ultrafilters on a measure separable Boolean
algebras which can be added by many random reals can be added also by a single
random real. This does not mean that the Stone space of such Boolean algebra
in the universe extended by adding many random reals is the same as in the
universe extended by a single random real. E.g. the Cantor algebra is measure
separable, as countable Boolean algebra, but the cardinality of its Stone
space clearly depends on how many random reals we add.

Notice that whenever $\mathbb{A}$ contains an uncountable independent
subfamily then it carries a measure which is non separable. Actually, under
$\mathsf{MA}_{\omega_{1}}$ the converse implication also holds true (see
[Fre97]) and so we have an immediate corollary.

######  Corollary 3.10.

Assume $\mathsf{MA}_{\omega_{1}}$. If $\mathbb{A}$ is a Boolean algebra
without an uncountable independent subfamily, then each ultrafilter on
$\mathbb{A}$ added by random reals can be added by a single random real.

It is well known that consistently there are Boolean algebras without an
uncountable independent family supporting non-separable measures. Thus, even a
small Boolean algebra may have $\mathbb{M}_{\omega_{1}}$-names for
ultrafilters which cannot be add by a single random real. In fact, under
$\mathsf{CH}$, there is a Boolean algebra with a much stronger property. The
following theorem was proved in [Kun81].

######  Theorem 3.11 (Kunen).

Assume $\mathsf{CH}$. There is a Boolean algebra $\mathbb{A}$ (with the Stone
space $K$) and a measure $\mu$ on $\mathbb{A}$ such that

  * •

$\mathrm{Bor}(K)_{/\hat{\mu}=0}$ is metrically isomorphic to
$\mathbb{M}_{\omega_{1}}$ (in particular, $\hat{\mu}_{|B}$ is non-separable
for every $\hat{\mu}$-positive $B\subseteq K$).

  * •

$F\subseteq K$ is nowhere dense if and only if $\hat{\mu}(F)=0$,

  * •

$\mathbb{A}$ does not contain an uncountable independent family.

######  Corollary 3.12.

Assume $\mathsf{CH}$. There is a Boolean algebra $\mathbb{A}$ which supports a
measure and such that each separable measure is supported on a nowhere dense
subset of its Stone space.

###### Proof.

Let $\mathbb{A}$, $K$, $\mu$ be as in Theorem 3.11. Let $\nu$ be a separable
measure carried by $\mathbb{A}$.

We will work in $K$. First, we will show that $\hat{\mu}\perp\hat{\nu}$. By
Jordan decomposition $\hat{\nu}=\hat{\nu_{0}}+\hat{\nu_{1}}$, where
$\hat{\nu_{0}}\perp\hat{\mu}$ and $\hat{\nu_{1}}\ll\hat{\mu}$. Since $\nu$ is
separable, both $\hat{\nu}_{0}$ and $\hat{\nu}_{1}$ are separable, too. By
Radon-Nikodym theorem $\hat{\nu}_{1}=\int f\ d\hat{\mu}$ for a measurable
function $f\colon K\to\mathbb{R}$. Suppose for the contradiction that
$\hat{\nu}_{1}(K)>0$. Then the function $f$ is not almost everywhere $0$ and
so there is a $\hat{\mu}$-positive $B\subseteq K$ and $\varepsilon>0$ such
that $f(x)>\varepsilon$ for each $x\in B$. But then
$\hat{\nu}_{1}(A)>\varepsilon\hat{\mu}(A)$ for each $A\subseteq B$ and so
$\hat{\nu_{1}}_{|B}$ is not separable. Thus, $\hat{\nu}_{1}$ is not separable,
a contradiction. So $\hat{\nu}_{1}=0$ and so $\hat{\mu}\perp\hat{\nu}$.

So, there is $F\subseteq K$ such that $\hat{\mu}(F)=0$ and $\hat{\nu}(F)=1$.
But then $F$ is nowhere dense and $\hat{\nu}$ is supported by $\overline{F}$
which is still nowhere dense. ∎

So, forcing with $\mathbb{M}$ adds ultrafilters on $\mathbb{A}$ which can be
quite precisely ’localized’: for each $\mathbb{M}$-name $\dot{\mathcal{U}}$
for an ultrafilter on $\mathbb{A}$ we can find a nowhere dense set $D\subseteq
K$ in the ground model such that $\Vdash_{\mathbb{M}}\dot{\mathcal{U}}\in D$.

##  4\. Non-trivial convergent sequences of homomorphisms

In this section we will deal with homomorphisms into measure algebras in
isolation with its forcing interpretations. First, we observe that we may use
the fact that measure algebras are metrizable in a natural way to define
notions of convergence of homomorphisms.

######  Definition 4.1.

Let $\mathbb{A}$ be a Boolean algebra and let $(\varphi_{n})$ be a sequence of
homomorphisms $\varphi_{n}\colon\mathbb{A}\to\mathbb{M}_{\kappa}$, where
$\kappa$ is a cardinal number. We say that

  * •

$(\varphi_{n})$ converges to $\varphi$ _pointwise_ if
$d_{\lambda_{\kappa}}(\varphi_{n}(A),\varphi(A))$ converges to $0$ for each
$A\in\mathbb{A}$.

  * •

$(\varphi_{n})$ converges to $\varphi$ _uniformly_ if for each $\varepsilon>0$
there is $N$ such that for each $n>N$ and each $A\in\mathbb{A}$ we have
$d_{\lambda_{\kappa}}(\varphi_{n}(A),\varphi(A))<\varepsilon.$

The following simple fact implies in particular that the Stone topology is a
pointwise convergence topology.

######  Proposition 4.2.

Let $(\varphi_{n})$ be a sequence of homomorphisms
$\varphi_{n}\colon\mathbb{A}\to\\{0,1\\}$. For each $n$ define an ultrafilter
$U_{n}$ on $\mathbb{A}$ by $U_{n}=\varphi^{-1}_{n}[\\{1\\}]$. Then

  * •

$(\varphi_{n})$ converges pointwise if and only if $(U_{n})$ converges (in the
Stone topology),

  * •

$(\varphi_{n})$ converges uniformly if and only if $(U_{n})$ converges
trivially.

To some extend this proposition can be adapted to a more general situation of
sequence of homomorphisms to measure algebras and convergence in the random
models (see [BNS21]).

Motivated by Proposition 4.2 we say that a sequence of homomorphisms into a
measure algebra _converges non-trivially_ if it converges pointwise but not
uniformly. We will show that for every infinite Boolean algebra $\mathbb{A}$
there are nontrivial convergent sequences into measure algebras. In the proof
we will treat separately the case when $\mathbb{A}$ contains the Cantor
algebra and when it is superatomic.

######  Proposition 4.3.

Let $\mathbb{A}$ be a Boolean algebra containing the Cantor subalgebra. Then
there is sequence $(\varphi_{n})$ of homomorphisms
$\varphi_{n}\colon\mathbb{A}\to\mathbb{M}$ convergent pointwise but not
uniformly.

###### Proof.

We will treat the elements of the Cantor algebra $\mathbb{C}$ (embedded in
$\mathbb{A}$) as clopen subsets of $2^{\omega}$ and the elements of
$\mathbb{M}$ as Borel subsets of $2^{\omega}$. So, we define
$\bar{\varphi}\colon\mathbb{C}\to\mathbb{M}$ by $\bar{\varphi}(C)=C$
understanding $C$ once as a clopen set, once as a representative of an element
of $\mathbb{M}$. By Sikorski’s Extension Theorem ([KMB89, Theorem 5.9]), since
$\mathbb{M}$ is complete, there isa homomorphism $\varphi$ extending
$\bar{\varphi}$.

For $n\in\omega$ define $f_{n}\colon 2^{\omega}\to 2^{\omega}$ by

$f_{n}(a)=a+_{2}e_{n}$

where $+_{2}$ denotes addition modulo $2$ and

$e_{n}(i)=\begin{cases}1&\text{for }i=n\\\ 0&\text{for }i\neq n.\end{cases}$

Now for $n\in\omega$ define $\psi_{n}\colon\mathbb{M}\to\mathbb{M}$ by
$\psi_{n}(B)=f_{n}[B]$ (again, identifying elements of $\mathbb{M}$ with its
Borel representatives). Finally, let $\varphi_{n}=\psi_{n}\circ\varphi$. We
claim that the sequence $(\varphi_{n})$ is the sought one.

First, denote $C_{n}=\\{x\in 2^{\omega}\colon x(n)=1\\}$ and notice that
$\varphi_{n}(C_{n})=C_{n}^{c}=\varphi(C_{n})^{c}$. Thus,
$d_{\lambda}(\varphi_{n+1}(C_{n}),\varphi_{n}(C_{n}))=1$ and so
$(\varphi_{n})$ does not converge uniformly.

To see that $(\varphi_{n})$ converges pointwise, fix $\varepsilon>0$ and
$A\in\mathbb{A}$. Let $C\in\mathbb{C}$ be such that $\lambda(C\triangle
A)<\varepsilon/2$. Notice that $\varphi_{n}(C)=C$ for $n$’s big enough, as
elements of $\mathbb{C}$ are finite Boolean combinations of $C_{n}$’s. Since
$\varphi_{n}$ is a metric isomorphism,
$\lambda(\varphi_{n}(A)\triangle\varphi_{n}(C))<\varepsilon/2$ and so
$\lambda(\varphi_{n}(A)\triangle C)<\varepsilon/2$ for $n$’s big enough.
Hence, $\lambda(\varphi_{n}(A)\triangle A)<\varepsilon$ for $n$’s big enough
and so $\varphi_{n}$ converges to identity. ∎

Now we will show that the conclusion of Proposition 4.3 holds true also for
Boolean algebras which does not contain the Cantor algebra. Those Boolean
algebras are exactly the superatomic Boolean algebras mentioned in the
previous section. The Stone spaces of superatomic algebras are scattered
spaces. It is well-known and widely used that scattered spaces contain
nontrivial convergent sequences. However, we will enclose a proof of it as it
is difficult to find it in the literature (at least in the easily accessible
one). The following proof is based on the proof contained in [Sob17].

######  Proposition 4.4.

If $K$ is an infinite scattered compact space then $K$ contains a nontrivial
convergent sequence.

###### Proof.

Let $K$ be an infinite scattered compact space. Let $F_{0}=K$ and let $x_{0}$
be an isolated point of $F_{0}$. Suppose that for $n\in\omega$ we have
constructed $F_{n}$ and $x_{n}\in F_{n}$ and define
$F_{n+1}=F_{n}\setminus\\{x_{n}\\}$ and $x_{n+1}$ as an isolated point of
$F_{n+1}$. Define

$Y=\\{x_{n}\colon n\in\omega\\}.$

Then $Y$ is an infinite discrete subset of $K$ and so it is not closed and
$\text{bd}Y\neq\emptyset$. Hence, there is an isolated point $x\in\text{bd}Y$.
We claim that $x$ is a limit of a sequence from $Y$.

Indeed, $\\{x\\}$ and $\text{bd}Y\setminus\\{x\\}$ are closed and thus there
are open, disjoint $U,V\subseteq K$ such that $\\{x\\}\subseteq U$ and
$\text{bd}Y\setminus\\{x\\}\subseteq V$. Notice that since $Y$ is discrete
$\overline{U\cap Y}=(U\cap Y)\cup\\{x\\}$. Enumerate $U\cap Y=\\{y_{n}\colon
n\in\omega\\}$.

We claim that $(y_{n})$ converges to $x$. Indeed, let $W$ be an open
neighbourhood of $x$ and denote $Z=U\cap Y\setminus W$. Then
$\overline{Z}\cap\mathrm{bd}Y=\emptyset$ and so, since $Y$ is discrete and
$Z\subseteq Y$, $Z$ is closed. But again, since $Z$ is discrete, it means that
$Z$ is finite and so almost all elements of $(y_{n})$ belongs to $W$. As $W$
was arbitrary, $(y_{n})$ converges to $x$. ∎

######  Proposition 4.5.

If $\mathbb{A}$ is an infinite superatomic algebra, then there is a sequence
$(\varphi_{n})$ of homomoprhisms $\varphi_{n}\colon\mathbb{A}\to\mathbb{M}$
convergent pointwise but not uniformly.

###### Proof.

It follows from the proposition above and Proposition 4.2 as ultrafilers on
$\mathbb{A}$ correspond to homomorphisms into the trivial algebra (which is of
course a subalgebra of $\mathbb{M}$). ∎

######  Theorem 4.6.

For every infinite Boolean algebra $\mathbb{A}$ there is a sequence of
homomorphisms $\varphi_{n}\colon\mathbb{A}\to\mathbb{M}$ which converges
nontrivially.

###### Proof.

Apply Proposition 4.3 and Proposition 4.5. ∎

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