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\title{Borel Ruziewicz problem}
\author{}
\date{November 10, 2026}
\subjclass[2020]{Primary 28C10, 37A15; Secondary 03E15, 43A07}
\keywords{Borel Ruziewicz problem, finitely additive measures, spectral gap, Borel dynamics}

\begin{document}
\begin{abstract}
We prove that normalized surface measure is the unique finitely additive
rotation-invariant probability on the Borel subsets of $S^n$, $n\geq2$.
We also prove that Lebesgue measure is the unique finitely additive
measure on bounded Borel subsets of $\R^d$, $d\geq3$, invariant under
orientation-preserving rigid motions and normalized on the unit cube.
The main step is the annihilation of Borel null sets. A spectral gap
first gives bounded divergence certificates for compact null sets,
chosen in a jointly Borel way. A shared random threshold then combines
these certificates to cover arbitrary null sets. For Euclidean space,
the construction is applied to families of spheres about two centers;
spherical averaging completes the identification of the measure.
\end{abstract}
\maketitle

\begin{center}
\begingroup
\setlength{\fboxsep}{9pt}
\fbox{\begin{minipage}{0.92\textwidth}
\small
\textbf{AI attribution.}
The results in this paper were obtained by AI agents, including
GPT-6-Astra and 6.1-sol (OpenAI), and Fable 5.2 (Anthropic).
Konstantin Slutsky organized the AI agents but claims no mathematical
contributions to the content of this paper.
\end{minipage}}
\endgroup
\end{center}

\section{Introduction}
Let $\lambda_n$ denote normalized surface measure on $S^n$. The classical
Ruziewicz problem asks whether finite additivity, nonnegativity,
normalization, and rotation invariance characterize $\lambda_n$ on the
Lebesgue $\sigma$-algebra. Margulis~\cite{Margulis} and
Sullivan~\cite{Sullivan} proved this for $n\geq4$, and
Drinfeld~\cite{Drinfeld} settled the cases $n=2,3$.

The corresponding question on Borel sets has an additional difficulty.
A finitely additive probability defined on the Borel $\sigma$-algebra
need not be given on all subsets of null sets, and an invariant
extension to the Lebesgue completion is not among the hypotheses.
Nor can one immediately regard it as a mean on $L^\infty(\lambda_n)$:
that passage requires the measure to vanish on Borel null sets.
Establishing this property is the principal task of the present paper.

We use the term \emph{charge} for a nonnegative finitely additive
function on a Borel algebra. Charges on compact spaces are normalized
to have total mass one unless stated otherwise. Our first result is
the following.

\begin{mainthm}\label{first:sphere}
For every $n\geq2$, normalized surface measure is the unique
$\SO(n+1)$-invariant probability charge on $\Bor(S^n)$.
The same conclusion holds under invariance by any prescribed finite
symmetric set $S\subseteq\SO(n+1)$ with a spectral gap on
$L^2_0(S^n,\lambda_n)$.
\end{mainthm}

Here a spectral gap means that, for some $\kappa>0$,
\begin{equation}\label{eq:first-gap}
 \max_{s\in S}\|u\circ s^{-1}-u\|_2
       \geq\kappa\|u-\lambda_n(u)\|_2
       \qquad(u\in L^2(\lambda_n)).
\end{equation}
Finite sets of rotations with this property exist in every dimension
$n\geq2$ by the classical work cited above. To make the argument more
self-contained, Appendix~\ref{sec:gap-appendix} gives an elementary
construction using quaternion counting, multiplicity estimates, and
averaging about two axes. This belongs to the established circle of
arithmetic approaches to spectral gaps; see
Gamburd--Jakobson--Sarnak~\cite{GJS} and Bourgain--Gamburd~\cite{BG}.

The Euclidean result likewise requires no reflections. Write
$\Bor_b(\R^d)$ for the algebra of bounded Borel sets and
$\mathcal L^d$ for Lebesgue measure.

\begin{mainthm}\label{first:euclid}
Let $d\geq3$. If $\mu:\Bor_b(\R^d)\to[0,\infty]$ is finitely
additive, invariant under $x\mapsto gx+v$ for $g\in\SO(d)$ and
$v\in\R^d$, and satisfies $\mu([0,1]^d)=1$, then
$\mu(A)=\mathcal L^d(A)$ for every bounded Borel set $A$.
\end{mainthm}

Margulis~\cite{MargulisEucl} proved the analogous uniqueness theorem
on bounded Lebesgue measurable sets. We give a direct proof of the
identification step once Borel null sets have been annihilated.

\subsection{Context}
Conley, Jackson, Marks, Seward, and Tucker-Drob~\cite{CJMST} explicitly
state the Borel Ruziewicz question, referring to Question~11.13 in
Wagon's book. They also construct a continuous action on another Polish
space which admits an invariant Borel probability charge but no
invariant countably additive probability. Thus the spherical theorem
requires a property beyond the mere absence of countably additive
invariant probabilities on a null part of the action.

Grabowski, M\'ath\'e, and Pikhurko~\cite{GMP} study the Borel question
in connection with measurable equidecompositions. Their uniqueness
theorem on sets which are both Lebesgue measurable and have the Baire
property, and the further proof by Mycielski and Tomkowicz~\cite{MT},
concern a larger algebra than the Borel sets. The existence of an
invariant extension to that algebra cannot be inferred from those
uniqueness statements. Unger's lectures~\cite{Unger} describe the
Borel problem and approaches through bounded flows.

The classical functional-analytic part of the argument is the
implication from spectral gap to uniqueness of an invariant mean on
$L^\infty$; see Rosenblatt~\cite{Rosenblatt}. Our proof uses that
principle only after null-set annihilation. All assertions about
divergence before that step hold at every point, rather than merely
almost everywhere.

\subsection{Outline}
Fix finitely many rotations with a spectral gap. Smoothing a singular
countably additive probability shows that some generator moves it by
a fixed amount. A separation argument then produces a bounded Borel
field whose divergence is almost one on any given compact null set.
The bound does not depend on that set. A countable search over finite
formulas makes the fields jointly Borel in a compact parameter.

For a general null set, choose closed covers with rapidly decreasing
measure. The word distance to the complement of a finite thickening
of each cover changes by at most one along a generator. Comparing this
distance with a shared random threshold selects a compact null set
containing the base point. Neighboring points usually select the same
compact set. Averaging the corresponding fields yields positive
divergence on a Borel set containing the original null set, and
nonnegative divergence everywhere else. Integration against an
invariant charge forces the set to have charge zero.

In Euclidean space, polar coordinates first provide suitable covers
outside an exceptional null set of radii. A second center converts
those exceptional radii into null spherical sections. The parameter
version of the same construction therefore annihilates every bounded
null set. Finally, three spherical averagings turn a bounded Borel
function of bounded support into a continuous compactly supported
function, where ordinary Haar uniqueness applies.

\subsection{Notation}
For a finite symmetric family $S$ of homeomorphisms, write
$\Gamma=\langle S\rangle$ and $\Gamma_r=\{g:|g|_S\leq r\}$.
The family may have repeated elements; its indices are paired by
an involution representing inversion. A field is a family of bounded
Borel functions $F_s$, with divergence
\[
                  D_SF(x)=\sum_{s\in S}
                       \bigl(F_s(x)-F_s(s^{-1}x)\bigr).
\]
If $\mu$ is invariant, then $\mu(D_SF)=0$. In particular a field
satisfying $D_SF\geq a\ind_N$ everywhere, with $a>0$, certifies
$\mu(N)=0$. We write $B_b(X)$ for bounded real Borel functions;
all its norms are pointwise supremum norms. Total variation is
normalized so that mutually singular probabilities have distance two.
We assume standard measure theory and functional analysis; our
descriptive-set-theoretic conventions follow \cite{Kechris}.

\section{A gap for singular probabilities}
Let $X$ be compact metric, $m$ a fully supported Borel probability,
and $S$ a finite nonempty symmetric family of $m$-preserving
isometries. Suppose \eqref{eq:first-gap} holds with $m$ in place of
$\lambda_n$.

\begin{proposition}\label{first:gap}
Set $c=\kappa^2/2$. For every countably additive Borel probability
$\nu\perp m$ there exist $s\in S$ and $g\in C(X,\R)$ such that
\begin{equation}\label{eq:first-singular}
              \|g\|_\infty\leq1,
              \qquad \int(g\circ s-g)\,d\nu\geq c.
\end{equation}
\end{proposition}

\begin{proof}
Let $a_r(x,y)=\max(0,1-d(x,y)/r)$ and normalize it in the second
coordinate:
\[
 p_r(x,y)=\frac{a_r(x,y)}{\int a_r(x,z)\,dm(z)},\qquad
                     h_r(y)=\int p_r(x,y)\,d\nu(x).
\]
The denominator is positive and bounded away from zero for each
$r>0$. The resulting smoothing preserves mass, commutes with the
generators, and has continuous densities. Its adjoint sends a
bounded Borel function of norm at most one to a continuous function
of norm at most one.

Suppose \eqref{eq:first-singular} fails. Symmetry of the test class
then gives $\|s_*(h_rm)-h_rm\|_{\TV}\leq c$ for every $s$.
For $u_r=\sqrt{h_r}$,
\[
 \|U_su_r-u_r\|_2^2\leq\|h_r\circ s^{-1}-h_r\|_1\leq c,
 \qquad \|u_r\|_2=1.
\]
The spectral gap implies $m(u_r)^2\geq1/2$.

On the other hand, singularity forces $m(u_r)\to0$. To see this,
choose a compact null set $K$ with $\nu(K)>1-\eta$, and an open
$U\supseteq K$ with $m(U)<\eta$. For sufficiently small $r$, the
smoothing of $\nu|_K$ is carried by $U$, so
$\int_{X\setminus U}h_r\,dm<\eta$. Cauchy--Schwarz on $U$ and
its complement gives $m(\sqrt{h_r})\leq2\sqrt\eta$.
This contradicts the preceding lower bound.
\end{proof}

In Sections~\ref{sec:first-sources}--\ref{sec:first-kernel} we only
use the conclusion \eqref{eq:first-singular}, with a fixed $c>0$.
The generators may there be homeomorphisms preserving $m$; they need
not be isometries.

\section{Compact certificates and Borel choice}\label{sec:first-sources}
For a compact set $K\subseteq X$, let $\E(K)$ consist of finite sums
\begin{equation}\label{eq:first-class}
                 \sum_{j=1}^a f_j\prod_{g\in W_j}\ind_{gK},
                 \qquad f_j\in C(X,\R),
\end{equation}
where $W_j$ is a finite list of group words. Empty products are
allowed, so $C(X)\subseteq\E(K)$. No support assertion outside
$\Gamma K$ is imposed on this class.

\begin{lemma}\label{first:source}
If $m(K)=0$, then for every $\varepsilon>0$ there is a field
$q\in\E(K)^S$ such that
\[
       \max_s\|q_s\|_\infty<2/c,
       \qquad D_Sq\geq\ind_K-\varepsilon\quad\text{everywhere}.
\]
\end{lemma}

\begin{proof}
Work in the Banach space of all bounded functions on $X$. Let $W$
be the downward closure of the divergences of fields in $\E(K)^S$
with component norms less than $2/c$. This set is convex. If the conclusion fails, $W$
is disjoint from the open $\varepsilon$-ball about $\ind_K$.
Separation gives a bounded linear functional $\Phi$ and $a\geq0$
such that
\begin{equation}\label{eq:first-separator}
          \Phi(D_Sq)\leq a<\Phi(\ind_K)
          \qquad\bigl(\max_s\|q_s\|_\infty<2/c\bigr).
\end{equation}
The functional is positive, since $W$ contains every negative ray.

Put $\Lambda_j=\Gamma_jK$ and $A_j=\Phi(\ind_{\Lambda_j})$.
Then $0<A_0\leq A_j\leq\Phi(1)$. For each $j$, Riesz
representation applied to $f\mapsto\Phi(f\ind_{\Lambda_j})$
gives a countably additive measure of mass $A_j$, carried by the
closed null set $\Lambda_j$. After normalization,
\eqref{eq:first-singular} supplies $s$ and $g$, $\|g\|_\infty\leq1$,
with
\begin{equation}\label{eq:first-cutdown-gap}
       cA_j\leq\Phi((g\circ s)\ind_{\Lambda_j})
                                      -\Phi(g\ind_{\Lambda_j}).
\end{equation}
The support statement follows directly from the cutdown: a
continuous function vanishing on $\Lambda_j$ has integral zero.

Use \eqref{eq:first-separator} with the single nonzero component
$-t g\ind_{s\Lambda_j}$ at index $s^{-1}$, where $t=3/(2c)<2/c$.
It gives
\[
       t\bigl(\Phi((g\circ s)\ind_{\Lambda_j})
                             -\Phi(g\ind_{s\Lambda_j})\bigr)\leq a.
\]
For $j=n+1$, both $\Lambda_j$ and $s\Lambda_j$ contain
$\Lambda_n$ and lie in $\Lambda_{n+2}$. Positivity therefore bounds
the change between their cutdowns by $A_{n+2}-A_n$.
Combining with \eqref{eq:first-cutdown-gap} yields
\[
         \frac3{2c}\bigl(cA_{n+1}-(A_{n+2}-A_n)\bigr)
                            \leq a<A_0\leq A_{n+1}.
\]
Thus $A_{n+2}-A_n>cA_0/3$. Iterating on the even indices contradicts
$A_j\leq\Phi(1)$.
\end{proof}

\begin{proposition}\label{first:selection}
Let $P$ be compact metrizable and $\mathcal K\subseteq P\times X$
closed. For every $\varepsilon>0$ there is a jointly Borel family
$F^p_s(x)$ such that
\[
 0\leq F^p_s(x)\leq4/c\quad(p\in P,s\in S,x\in X),
 \qquad D_SF^p(x)\geq1-\varepsilon
        \quad(x\in\mathcal K_p, m(\mathcal K_p)=0).
\]
\end{proposition}

\begin{proof}
Fix a countable dense sequence in $C(X,\R)$ and enumerate fields
of the form \eqref{eq:first-class} with coefficients in that
sequence. A code is valid at $p$ when its component norms are at
most $2/c$ and its divergence is at least $1-\varepsilon$ on
$\mathcal K_p$.

For a fixed code, these conditions are Borel in $p$. Indeed, the
translated membership sets in $P\times X$, including the pullbacks
appearing in divergence, are closed. Their finite
Boolean combinations partition the product into $\sigma$-compact
pieces on which the code evaluations and their divergences are
continuous. The violation set for either a component bound or the
divergence inequality is therefore $\sigma$-compact. Its projection
to $P$ is $\sigma$-compact, hence Borel. The component bounds are
tested on all of $P\times X$, and the divergence condition only
on $\mathcal K$.

Lemma~\ref{first:source}, used with $\varepsilon/2$, followed by
sufficiently close approximation of its finitely many continuous
coefficients, gives a valid code whenever the fiber is null.
Choose the least valid code, and choose zero when none exists.
This is a jointly Borel field $q^p$, bounded by $2/c$ in each
component. Reorient it by
\[
                F^p_s=(q^p_s)^++(q^p_{s^{-1}})^-\circ s.
\]
This preserves divergence and gives the asserted nonnegative bound.
\end{proof}

\section{The random-threshold kernel}\label{sec:first-kernel}

\begin{lemma}\label{first:cover}
Let $\nu$ be a finite Borel measure on a compact metric space.
If $\nu^*(N)=0$ and $\eta_r>0$, there are closed sets $C_r$ with
$\nu(C_r)\leq\eta_r$ and $N\subseteq\limsup_r C_r$.
\end{lemma}
\begin{proof}
Choose open $O_j\supseteq N$ with $\nu(O_j)<2^{-j}$. Closed
shells defined by the intervals $[1,\infty)$ and
$[1/(i+1),1/i]$ in the distance to the complement cover $O_j$
with multiplicity at most two. If $O_j$ is the whole space use it
as a single closed set. Enumerating all shells gives a sequence
$(D_a)$ of summable measures whose limsup contains $N$. Group it
into successive finite blocks, starting each block far enough out
that the remaining measure sum is at most $\eta_r$.
\end{proof}

\begin{proposition}\label{first:kernel}
Let $C_r\subseteq X$ be closed, and let $L_r$ be positive integers
such that $r\leq L_r$ and
\[
               \sum_{r\geq0}L_r^{-1}\leq\frac{c}{16|S|}.
\]
If $m(\Gamma_{2L_r}C_r)\to0$, then $\limsup_r C_r$ is contained
in a Borel set $Z$ carrying a bounded Borel field $H$ with
$D_SH\geq\tfrac12\ind_Z$ everywhere.
\end{proposition}

\begin{proof}
Put $E_r=\Gamma_{2L_r}C_r$ and let
\[
 t_r(x)=\min\bigl(2L_r,\min\{|g|:gx\notin E_r\}\bigr),
\]
with $\min\emptyset=\infty$. Its superlevel sets are closed,
$|t_r(sx)-t_r(x)|\leq1$, and $t_r=2L_r$ on $C_r$.
Define
\[
 Z=\{x:\forall n\ \exists r\geq n, t_r(x)\geq L_r+n\}.
\]
This set is Borel and invariant. It contains $\limsup C_r$, since
$L_r\geq r$.

For $\omega\in[0,1)$ set
\[
 \theta_r(\omega)=\lfloor L_r\omega\rfloor,
 \qquad b_r(x,\omega)=\ind\{\theta_r(\omega)<t_r(x)\}.
\]
Use the compact parameter space $P=\{0,1\}^{\N}$ and the closed
family
\[
               \mathcal K=\{(b,y):b_r=1\Rightarrow y\in E_r
                                           \text{ for every }r\}.
\]
For every $x$, its selected fiber contains $x$. If $x\in Z$, then
$b_r(x,\omega)=1$ for infinitely many $r$, so
$m(\mathcal K_{b(x,\omega)})=0$ by $m(E_r)\to0$.

Apply Proposition~\ref{first:selection} with $\varepsilon=1/4$.
Define
\[
 H_s(x)=\int_0^1 F_s^{b(x,\omega)}(x)\,d\omega\quad(x\in Z),
 \qquad H_s(x)=0\quad(x\notin Z).
\]
All parameter maps are Borel, so these functions are Borel and
bounded. Fix $x\in Z$. Using the same $\omega$ at $x$ and
$s^{-1}x$, the integrand of its divergence is
\[
 D_SF^{b(x,\omega)}(x)+
 \sum_s\bigl(F_s^{b(x,\omega)}(s^{-1}x)
                   -F_s^{b(s^{-1}x,\omega)}(s^{-1}x)\bigr).
\]
The first term is at least $3/4$. A summand vanishes when the
parameters agree and is otherwise at least $-4/c$.
At coordinate $r$ the parameters can differ only when
$\theta_r$ equals the smaller of the two neighboring distances,
an event of measure at most $L_r^{-1}$. The union bound gives
\[
                   D_SH(x)\geq\frac34-\frac{4|S|}{c}
                                      \sum_rL_r^{-1}\geq\frac12.
\]
Invariance of $Z$ makes the divergence zero outside $Z$.
No integration against a finitely additive charge has been used
in this construction.
\end{proof}

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[x=1.2cm,y=.6cm,font=\small]
 \node[anchor=east] at (-.6,1.5) {$\theta_r$};
 \node[anchor=east] at (-.6,.5) {$t_r=2$};
 \node[anchor=east] at (-.6,-.5) {$t_r=3$};
 \foreach \w in {0,1,2,3}{
  \node at (\w,1.5) {$\w$};
  \draw[fill=blue!5] (\w-.4,.1) rectangle (\w+.4,.9);
  \draw[fill=blue!5] (\w-.4,-.9) rectangle (\w+.4,-.1);
 }
 \foreach \w in {0,1}{\node at (\w,.5) {$1$};\node at (\w,-.5) {$1$};}
 \node at (2,.5) {$0$};\node at (3,.5) {$0$};
 \draw[fill=red!10] (1.6,-.9) rectangle (2.4,-.1);
 \node at (2,-.5) {$1$};\node at (3,-.5) {$0$};
\end{tikzpicture}
\caption{The selected bit $\ind\{\theta_r<t_r\}$ for $L_r=4$.
Neighboring distances disagree for only one threshold. The same
random number $\omega$ is used in every coordinate and at every
base point. Independence between coordinates is not required.}
\end{figure}

\begin{theorem}\label{first:null}
Every invariant probability charge on $X$ annihilates every
$m$-null Borel set.
\end{theorem}
\begin{proof}
Choose $L_r=N_0 2^r$ with $N_0$ large enough for
Proposition~\ref{first:kernel}. Since
$|\Gamma_a|\leq(|S|+1)^a$, Lemma~\ref{first:cover} provides
$C_r$ whose limsup contains the given null set and such that
$m(\Gamma_{2L_r}C_r)\leq1/(r+1)$. The kernel proposition gives
$D_SH\geq\ind_Z/2$. Invariance and positivity imply
$0=\mu(D_SH)\geq\mu(Z)/2$.
\end{proof}

\section{Invariant means and the spherical theorem}
We record the parameter version of the classical uniqueness argument,
since it will be used again for Euclidean space. Let $(X,m)$ be
compact metric with a probability-preserving finite nonempty symmetric family $S$
satisfying \eqref{eq:first-gap}. Let $T$ be compact metric and
$\alpha$ a finite Borel measure on $T$. The lifted action fixes $T$.

\begin{lemma}\label{first:fibreMean}
Let $\varphi$ be an invariant mean on bounded Borel functions on
$T\times X$ which annihilates $(\alpha\otimes m)$-null Borel sets.
For every bounded Borel $f$,
\[
       \varphi(f)=\varphi(Ef),\qquad
                        Ef(t,x)=\int_X f(t,y)\,dm(y).
\]
\end{lemma}
\begin{proof}
Normalize $\alpha$, the zero-measure case being incompatible with
$\varphi(1)=1$. Null-set annihilation lets us work in $L^\infty$.
Probability densities are weak-$*$ dense in its positive dual unit
sphere. For a fixed $f$, separation and the equality of weak and
norm closures of convex sets in $(L^1)^S\times\R$ give densities
$w_j$ with
\[
 \sum_s\|U_sw_j-w_j\|_1\to0,
 \qquad\int(f-Ef)w_j\to\varphi(f-Ef).
\]
Set $u_j=\sqrt{w_j}$ and $a_j=Eu_j$. Applying the fiberwise gap
in sum form and integrating in $t$ gives
\[
 \kappa^2\|u_j-a_j\|_2^2
 \leq\sum_s\|U_su_j-u_j\|_2^2
 \leq\sum_s\|U_sw_j-w_j\|_1\to0.
\]
Since $\|u_j\|_2=1$ and $\|a_j\|_2\leq1$,
$\|w_j-Ew_j\|_1\leq2\|w_j-a_j^2\|_1\leq4\|u_j-a_j\|_2$.
Self-adjointness of $E$ now gives
$\int(f-Ef)w_j=\int f(w_j-Ew_j)\to0$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{first:sphere}]
For a prescribed finite spectral-gap family,
Proposition~\ref{first:gap} and Theorem~\ref{first:null} annihilate
Borel null sets. The charge therefore descends to an invariant
mean on $L^\infty(\lambda_n)$. Lemma~\ref{first:fibreMean}, with
a one-point parameter space, identifies it with the spherical
integral. Existence of a spectral-gap family, established in
Appendix~\ref{sec:gap-appendix}, gives the assertion for all rotations.
\end{proof}

\section{Euclidean space}

\subsection{Rapid fiberwise covers}
Keep $X,m,S$ as in the preceding section, and assume in addition
the continuous-test singular gap \eqref{eq:first-singular} for a fixed
$c>0$. For compact metric $T$
and closed $C_r\subseteq T\times X$, write
$C_r(t)=\{x:(t,x)\in C_r\}$. Say these sets give a \emph{rapid
fiberwise cover} of $N\subseteq T\times X$ if
\begin{equation}\label{eq:first-rapid}
 N\subseteq\limsup_r C_r,
 \qquad A^{2^r}m(C_r(t))\longrightarrow0
 \quad\text{for every }(t,x)\in N\text{ and every integer }A\geq1.
\end{equation}

\begin{proposition}\label{first:rapid}
Every invariant probability charge for the lifted action on
$T\times X$ annihilates a Borel set admitting a rapid fiberwise
cover.
\end{proposition}
\begin{proof}
Choose $L_r=N_0 2^r$ as before, and let $G$ be the Borel set of
parameters for which the decay condition in \eqref{eq:first-rapid}
holds. The lifted thickenings satisfy
\[
 m((\Gamma_{2L_r}C_r)(t))
       \leq(|S|+1)^{2L_r}m(C_r(t))\longrightarrow0
                       \qquad(t\in G).
\]
Use the construction of Proposition~\ref{first:kernel} with compact
parameter space $\{0,1\}^{\N}\times T$. Its compact fibers are
\[
                  \bigcap_{b_r=1}(\Gamma_{2L_r}C_r)(t),
\]
and their full membership family is closed. For kernel points over $G$, the selected codes have infinitely many
active coordinates, so their compact fibers are null. The same countable disagreement estimate gives a
bounded Borel certificate on the invariant part of the kernel set
over $G$; extend by zero elsewhere. It contains $N$, and integrating
the certificate proves the assertion.
\end{proof}

\begin{lemma}\label{first:fubini}
Let $\alpha$ be a finite Borel measure on compact metric $T$.
Every $(\alpha\otimes m)$-null set $N\subseteq T\times X$ has a
closed limsup cover $(C_r)$ satisfying \eqref{eq:first-rapid}
over an $\alpha$-conull Borel set of parameters.
\end{lemma}
\begin{proof}
By Lemma~\ref{first:cover}, choose $C_r$ with
$(\alpha\otimes m)(C_r)\leq2^{-r}(r+1)^{-2^r}$.
Tonelli shows that
\[
                \sum_r(r+1)^{2^r}m(C_r(t))<\infty
\]
outside an $\alpha$-null Borel set. Its summands tend to zero,
and each fixed $A$ is eventually at most $r+1$.
\end{proof}

\subsection{Two centers}
Fix $d\geq3$, put $X=S^{d-1}$, and choose spectral-gap rotations
$S\subseteq\SO(d)$. For a center $p$ and $R>0$, use the polar map
$\pi_p:[0,R]\times S^{d-1}\to\R^d$,
$\pi_p(t,\xi)=p+t\xi$. We say a set has a rapid cover about $p$
if its polar preimage has a cover as in \eqref{eq:first-rapid}.

\begin{proposition}\label{first:twoCentre}
Let $p\ne q$ and let $N\subseteq\bar B(p,R)\cap\bar B(q,R)$
be Borel and Lebesgue-null, with $p,q\notin N$. Then
$N=N_1\cup N_2$ for Borel sets $N_i$ having rapid covers about
$p$ and $q$, respectively.
\end{proposition}
\begin{proof}
Polar coordinates make $\pi_p^{-1}(N)$ null for the product of
Lebesgue measure on $[0,R]$ and spherical probability. Apply
Lemma~\ref{first:fubini}, and denote its bad radii by
$B\subseteq[0,R]$. Put
\[
               N_2=N\cap\{|x-p|\in B\},\qquad N_1=N\setminus N_2.
\]
The first set of good fibers gives the required cover of $N_1$.

The set $B^{(2)}=\{t^2:t\in B\}$ is null. Choose closed
$J_r\subseteq[0,R^2]$ with
$B^{(2)}\subseteq\limsup J_r$ and
$|J_r|\leq2^{-r}(r+1)^{-2^r}$. The closed sets
\[
          C'_r=\{(t,\xi):|q+t\xi-p|^2\in J_r\}
\]
cover the polar preimage of $N_2$ infinitely often. Let
$\ell=|q-p|$ and $e=(q-p)/\ell$. For $t>0$,
\[
       |q+t\xi-p|^2=t^2+\ell^2+2t\ell\langle\xi,e\rangle.
\]
The coordinate $\langle\xi,e\rangle$ on $S^{d-1}$ has density
proportional to $(1-u^2)^{(d-3)/2}$ on $(-1,1)$, bounded by a
constant $M_d$. Consequently
\[
                 \lambda_{d-1}(C'_r(t))
                       \leq\frac{M_d}{2t\ell}|J_r|.
\]
For each fixed positive $t$ this implies the required rapid decay.
The fiber $t=0$ contains no point of the polar preimage of $N_2$,
because $q\notin N$.
\end{proof}

Let $\mu$ satisfy the assumptions of Theorem~\ref{first:euclid}.
It is finite on bounded sets, since finitely many translates of the
unit cube cover any such set. Singletons have charge zero by placing
arbitrarily many distinct translates inside a fixed cube. For a
bounded null Borel $N$, remove two centers, choose $R$, and apply
Proposition~\ref{first:twoCentre}. The restriction of the polar map to $(0,R]\times S^{d-1}$
is a Borel bijection onto the punctured closed ball intertwining the lifted generators with
rotations about the center. If the ball has positive charge,
normalize its restriction and transport it to the polar product,
extending by zero on the zero-radius fiber;
if it has zero charge the conclusion is immediate. Proposition~
\ref{first:rapid} gives
\begin{equation}\label{eq:first-euclid-null}
          \mu(N)=0\qquad(N\text{ bounded, Borel, and Lebesgue-null}).
\end{equation}

\subsection{Spherical averaging}
Let $L$ denote integration against $\mu$ on bounded Borel functions
of bounded support. For $p\in\R^d$ put
\[
              E_pf(x)=\int_{S^{d-1}}f(p+|x-p|\xi)\,d\lambda_{d-1}(\xi).
\]
Spherical averaging preserves boundedness, Borel measurability,
bounded support, and the Lebesgue integral.

\begin{lemma}\label{first:avg}
For every bounded Borel function $f$ of bounded support,
$L(f)=L(E_pf)$.
\end{lemma}
\begin{proof}
Work in a ball about $p$ containing both supports. When its charge
is positive, normalize the restriction and pass to polar coordinates.
By \eqref{eq:first-euclid-null}, the resulting invariant mean
annihilates null sets for the product of radial Lebesgue measure
and spherical measure. Lemma~\ref{first:fibreMean} gives the
identity. The center is harmless because its charge is zero.
This argument does not interchange integration against the charge
with integration over the sphere.
\end{proof}

\begin{lemma}\label{first:smoothing}
If $p\ne q$ and $f$ is bounded Borel with bounded support, then
$E_pE_qE_pf$ is continuous and has compact support.
\end{lemma}
\begin{proof}
Write $E_pf(x)=h(|x-p|^2)$ with $h$ bounded Borel. For
$\ell=|q-p|$, the second average at distance $t>0$ from $q$ is
\[
             b(t)=\int_{-1}^1
                 h(t^2+\ell^2+2t\ell u)g_d(u)\,du,
\]
where $g_d$ is the bounded coordinate density used above. Under
the affine substitution $v=t^2+\ell^2+2t\ell u$, its density
varies continuously in $L^1$ for $t>0$. This follows by dominated
convergence: in a neighborhood of a positive $t$, the transformed
densities have uniformly bounded supports and bounds and converge
pointwise away from the endpoints. Hence $E_qE_pf$ is continuous
off $q$.

If a bounded Borel $F$ is continuous off $q$, then $E_pF$ is
continuous everywhere for $p\ne q$. For $x_j\to x$ the evaluation
points $p+|x_j-p|\xi$ converge; the limit equals $q$ for at most
one $\xi$ when $x\ne p$, and for no $\xi$ when $x=p$. Spherical
measure is atomless, so dominated convergence applies. Bounded
support is preserved at each averaging; its closure is compact
in finite-dimensional Euclidean space.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{first:euclid}]
On $C_c(\R^d)$, the positive translation-invariant functional $L$
is $a\mathcal L^d$ for some $a\geq0$, by Riesz representation and
uniqueness of Haar measure. Given bounded Borel $A$, apply the
preceding lemmas to $f=\ind_A$ and $h=E_pE_qE_pf$, with $p\ne q$.
Then
\[
                 \mu(A)=L(f)=L(h)=a\int h=a\mathcal L^d(A).
\]
The unit-cube normalization gives $a=1$.
\end{proof}

\appendix
\section{An elementary spectral-gap construction}\label{sec:gap-appendix}
We give an elementary construction of the finite rotation families
used in Theorem~\ref{first:sphere}. The proof starts on $S^3$, viewed
as the group of unit quaternions, transfers to $S^2$, and then raises
the dimension. The arithmetic and multiplicity arguments are in the
spirit of \cite{GJS,BG}. The existence theorem is classical; our
purpose here is to supply the analytic input within the paper.

\subsection{Quaternion words}
Identify $G=S^3$ with the unit quaternions and let $m$ be Haar
probability. The units satisfy $i^2=j^2=k^2=ijk=-1$, and
$\bar q$ denotes quaternionic conjugation. Put
\[
 \mathcal A=\{1\pm2i,1\pm2j,1\pm2k\},\qquad
 S=\{\alpha/\sqrt5:\alpha\in\mathcal A\},\qquad
 Tf(x)=\tfrac16\sum_{s\in S}f(s^{-1}x).
\]
The quaternion norm is $N(a+bi+cj+dk)=a^2+b^2+c^2+d^2$.
Left multiplication by a unit quaternion is a rotation of $\R^4$.
For $l\geq1$, let $p_l(\eta)$ be the proportion of words of length
$2l$ in $S$ whose product is within Euclidean distance $\eta$ of $1$.
Words here are counted with multiplicity, without reduction.

\begin{lemma}[Nonconcentration]\label{lem:quaternion-count}
There is a constant $C$ such that, for $l\geq1$ and $0<\eta\leq1$,
\begin{equation}\label{eq:nonconcentration}
                 p_l(\eta)\leq(3/4)^{2l}+C\eta^3 5^{l/20}.
\end{equation}
\end{lemma}

\begin{proof}
We first record the elementary estimate
\begin{equation}\label{eq:three-squares}
 r_3(n):=|\{(a,b,c)\in\Z^3:a^2+b^2+c^2=n\}|
                         \leq C_0 n^{1/2+1/40}\qquad(n\geq1).
\end{equation}
Write $r_2(t)$ for the number of representations $t=a^2+b^2$ with
$a,b\in\Z$. Then $r_3(n)\leq(2\sqrt n+1)\max_{0\leq t\leq n}r_2(t)$.
For $t>0$, primitive representations $a^2+b^2=t$ map to square
roots of $-1$ modulo $t$ by $ab^{-1}$, with at most four
representations per root. To see the fiber bound, if two primitive
pairs give the same root, both $aa'+bb'$ and $ab'-a'b$ are divisible
by $t$, while the sum of their squares is $t^2$; the pairs therefore
differ by one of the four Gaussian units. The Chinese remainder
theorem bounds the number of roots by $2^{\omega(t)}\leq d(t)$,
where $\omega(t)$ counts distinct prime factors and $d(t)$ counts
positive divisors.
Summing over common divisors gives $r_2(t)\leq4d(t)^2$; also $r_2(0)=1$.
Finally $d(t)\leq C_\varepsilon t^\varepsilon$: in
$d(t)/t^\varepsilon=\prod_p(e_p+1)p^{-\varepsilon e_p}$ the factors
are at most one for all sufficiently large primes, and uniformly
bounded for each of the finitely many remaining primes. Take
$\varepsilon=1/80$ to obtain \eqref{eq:three-squares}.

For an integral quaternion $\beta$ of norm $5^L$, write
$\beta=5^v\gamma$ with $\gamma$ not divisible by $5$, and define
its level to be $j=L-2v$. Let $N_L(\beta)$ count length-$L$ words
in $\mathcal A$ with product $\beta$. The following elementary
divisibility observation controls multiplicities: for primitive
$\gamma$, at most one $\alpha\in\mathcal A$ has
$5\mid\gamma\bar\alpha$. Otherwise $5$ divides
$\gamma(\bar\alpha-\bar\alpha')$, and multiplication by its
conjugate forces $5\mid\gamma$, since
$N(\bar\alpha-\bar\alpha')\in\{8,16\}$. If
$\gamma\bar\alpha=5\gamma'$, the quaternion $\gamma'$ is also
primitive: multiplying by $\alpha$ would otherwise make $\gamma$
divisible by $5$.

Define nonnegative numbers $a_L(j)$ by
\[
 a_0(j)=\ind_{\{0\}}(j),\qquad
 a_{L+1}(0)=6a_L(1),\qquad
 a_{L+1}(j)=a_L(j-1)+5a_L(j+1)\quad(j\geq1).
\]
They are the numbers of length-$L$ walks to a fixed vertex at
distance $j$ in the $6$-regular tree. Direct induction gives
\begin{equation}\label{eq:radial-walk}
 \begin{split}
 &a_L(j+2)\leq a_L(j),\qquad a_L(j)2^j\leq(9/2)^L,\\
 &a_L(0)+\sum_{j\geq1}6\cdot5^{j-1}a_L(j)=6^L.
 \end{split}
\end{equation}
For the middle bound, multiplying the recursion by $2^j$ gives
coefficients $2$ and $5/2$ away from zero, and coefficient $3$ at
zero. The final identity counts all walks by their endpoint.

Splitting a word by its last letter gives
\[
       N_{L+1}(\beta)=\sum_{\alpha:\,5\mid\beta\bar\alpha}
                                         N_L(\beta\bar\alpha/5).
\]
At positive level, there is at most one predecessor of level $j-1$;
all others have level $j+1$. If there is none of level $j-1$, the
first inequality in \eqref{eq:radial-walk} bounds six higher-level
terms by one lower-level term and five higher-level terms. At level
zero all predecessors have level one. Thus
\begin{equation}\label{eq:multiplicity-count}
                          N_L(\beta)\leq a_L(j).
\end{equation}

Take $L=2l$. At level zero, a product near $5^l$ has primitive part
$1$, so its contribution is at most $(9/2)^{2l}$. At level $2j>0$,
write $P=5^j$ and $\Rea\gamma=P-t$. Nearness to $1$ implies
\[
         1\leq t<\eta^2P/2,
         \qquad N(\operatorname{Im}\gamma)=t(2P-t)<\eta^2P^2.
\]
The lower bound $t\geq1$ follows from primitivity, since $P$ itself
is divisible by $5$. By \eqref{eq:three-squares}, the number of
possible primitive parts is at most
\[
 C_1\eta^2P(\eta^2P^2)^{1/2+1/40}
                         \leq C_1\eta^3 P^2 5^{l/20}.
\]
Each has word multiplicity at most $a_{2l}(2j)$. Since
$\sum_{j\geq1}5^{2j}a_{2l}(2j)\leq6^{2l}$, summation and division
by $6^{2l}$ prove \eqref{eq:nonconcentration}.
\end{proof}

\subsection{Multiplicity and smoothing on polynomial spaces}
Let $\Pol_k$ be the restrictions to $S^3$ of real polynomials of
degree at most $k$, and put $W_k=\Pol_k\ominus\Pol_{k-1}$, with
$W_0$ the constants. These spaces are finite dimensional, mutually
orthogonal, invariant under all rotations, and have dense algebraic
sum in $L^2(G,m)$. Write $R_qf(x)=f(xq)$.

\begin{lemma}\label{lem:representation-multiplicity}
If $k\geq1$ and $0\ne F\subseteq W_k$ is invariant under every
$R_q$, then $\dim F\geq(k+1)/2$.
\end{lemma}
\begin{proof}
For an orthonormal basis $e_1,\ldots,e_d$ of $F$, let
$K(x,y)=\sum_ae_a(x)e_a(y)$ and
$\Psi(z)=\int K(xz,x)\,dm(x)$. Right invariance of $F$ gives
$K(xq,yq)=K(x,y)$, hence $\Psi$ is a class function.
It has degree at most $k$, $\Psi(1)=d$, and is orthogonal to
$\Pol_{k-1}$: after $y=xz$, the inner integral against a polynomial
$p(z)$ becomes the pairing of $e_a(y)$ with $p(x^{-1}y)$.

Every unit quaternion is conjugate to $\cos\theta+i\sin\theta$.
The restriction of $\Psi$ to this circle is an even trigonometric
polynomial, so $\Psi(z)=Q(\Rea z)$ with $\deg Q\leq k$.
The real coordinate of Haar measure has density
$(2/\pi)\sqrt{1-t^2}$ on $(-1,1)$. Orthogonality therefore gives
$Q=cU_k$, where
$U_k(\cos\theta)\sin\theta=\sin((k+1)\theta)$ is the Chebyshev
polynomial of the second kind. Since $U_k(1)=k+1$, $c=d/(k+1)>0$.

On $F$ the operator
\[
              P_k=\frac1\pi\int_0^{2\pi}\cos(k\theta)
                           R_{\cos\theta+i\sin\theta}\,d\theta
\]
is an idempotent, by the group law and orthogonality of trigonometric
functions. Its trace is
$\pi^{-1}\int\cos(k\theta)cU_k(\cos\theta)\,d\theta=2c$.
This is a positive integer, being the rank of an idempotent. Hence
$2d/(k+1)\geq1$.
\end{proof}

For $f\in\Pol_k$ and a unit quaternion $q$, the restriction of $f$
to each great circle is a trigonometric polynomial of degree at most
$k$. The usual Fourier shift estimate, averaged over right
translations, yields
\begin{equation}\label{eq:bernstein}
            \|R_qf-f\|_2\leq\frac{\pi k}{2}\,\|q-1\|\,\|f\|_2.
\end{equation}
Indeed, if $q=\cos\alpha+u\sin\alpha$ with $0\leq\alpha\leq\pi$,
the shift bound is $k\alpha$ and
$\alpha\leq(\pi/2)\|q-1\|$.

\begin{lemma}\label{lem:HS}
If $F\subseteq\Pol_k$, $k\geq1$, is an eigenspace of $T$ with
eigenvalue $\lambda$, then
\begin{equation}\label{eq:HS}
              (\dim F)\lambda^{2l}
              \leq16\pi(\pi k)^3p_l(2/(\pi k)).
\end{equation}
\end{lemma}
\begin{proof}
Set $a=1/(\pi k)$, $B=\{q:\|q-1\|<a\}$,
$\Phi=\ind_B/m(B)$, and $Af=\int R_qf\,\Phi(q)\,dm(q)$.
By \eqref{eq:bernstein}, $\|Af\|_2\geq\|f\|_2/2$ on $\Pol_k$.
The coordinate density above gives $m(B)\geq a^3/(4\pi)$.

Left and right translations commute. For an orthonormal basis
$(e_j)$ of $F$, this gives
\[
 \lambda^{2l}\dim F/4
       \leq\sum_j\|T^lAe_j\|_2^2\leq\|T^lA\|_{\mathrm{HS}}^2.
\]
No invariance of $F$ under $A$ is needed in this inequality.
The integral kernel of $T^lA$ is
\[
                    6^{-l}\sum_{w\in S^l}\Phi(x^{-1}wy).
\]
Expanding its squared norm, the overlap for words $w,w'$ is at
most $m(B)^{-1}$, and vanishes unless
$\|w'w^{-1}-1\|<2a$. This follows after the substitution
$z=x^{-1}wy$, since two points of $B$ have quotient within $2a$
of the identity and the distance is invariant under conjugation.
Pairs of length-$l$ words correspond bijectively, as index sequences,
to length-$2l$ words by reversing and inverting one half. Thus
$\|T^lA\|_{\mathrm{HS}}^2\leq m(B)^{-1}p_l(2a)$, which proves
\eqref{eq:HS}.
\end{proof}

\begin{proposition}\label{prop:S3gap}
The six rotations given by left multiplication by $S$ have a
spectral gap on $L^2_0(S^3,m)$.
\end{proposition}
\begin{proof}
The operator $T$ is self-adjoint and preserves every $W_k$. Let
$\lambda_k$ be its largest eigenvalue there, ignoring zero spaces.
The corresponding eigenspace is invariant under all right
translations. Lemmas~\ref{lem:representation-multiplicity},
\ref{lem:HS}, and \ref{lem:quaternion-count} give
\[
       \lambda_k^{2l}\leq C_2 k^2(9/16)^l+C_3 k^{-1}5^{l/20}.
\]
For positive $\lambda_k$ and $l=\lceil6\log k\rceil$, the right
side is $O(k^{-1/2})$. For all sufficiently large $k\geq3$, it is
at most $k^{-1/4}$; since $l\leq7\log k$, this gives
$\lambda_k\leq e^{-1/56}<1$. Nonpositive eigenvalues require no
separate estimate.

For each remaining $k$, $\lambda_k<1$. Indeed, equality would give
a nonzero $f\in W_k$ fixed by every generator. The powers of
$(1+2i)/\sqrt5$ are dense in the $i$-circle: its order is infinite,
as the imaginary coordinate of $(1+2i)^r$ cycles through nonzero
values modulo $5$. The same holds for the $j$-circle, and these
two circles generate $\SU(2)$ by Euler angles. Thus a continuous
invariant function is constant, contradicting $f\in W_k$, $k\geq1$.

We have obtained $\sup_{k\geq1}\lambda_k<1$. Orthogonality of the
$W_k$, polynomial density, and
\[
        \sum_{s\in S}\|U_sf-f\|_2^2
                 =12\bigl(\|f\|_2^2-\langle Tf,f\rangle\bigr)
\]
give the required gap on all of $L^2_0$.
\end{proof}

\subsection{Transfer and the two-axis argument}
Conjugation of unit imaginary quaternions gives the usual map
$\SU(2)\to\SO(3)$. Pullback along its orbit map embeds
$L^2(S^2)$ equivariantly and isometrically in $L^2(S^3)$. Thus
Proposition~\ref{prop:S3gap} gives a spectral gap on $S^2$ as well.

\begin{proposition}\label{prop:dimension-gap}
If $S^n$, $n\geq2$, admits a finite spectral-gap rotation set, then
so does $S^{n+1}$.
\end{proposition}
\begin{proof}
Extend the rotations by the identity on the last coordinate, and
adjoin their conjugates by the orthogonal map interchanging the
first and last coordinates. All resulting maps have determinant one.
Let $P_1$ and $P_2$ be averaging over latitudes about these two axes.
The fiberwise spectral gap bounds
$\|u-P_1u\|_2^2+\|u-P_2u\|_2^2$ by a constant times the sum of
the new generator displacements. We show that these two deviations
control variance.

Write $\rho$ for the coordinate law on $S^n$, and set
\[
 K\phi(t)=\int\phi(\sqrt{1-t^2}\,s)\,d\rho(s),\qquad -1\leq t\leq1.
\]
The density of $\rho$ is proportional to
$(1-s^2)^{(n-2)/2}$ and is at least some $b>0$ on $[-1/2,1/2]$.
For $\phi\geq0$, restricting the two coordinates in $K^2$ to that
interval gives
\begin{equation}\label{eq:doeblin}
 K^2\phi(t)\geq b^2\int_{-1/4}^{1/4}\phi(v)\,dv.
\end{equation}
To check the uniform bound, the scale of the inner coordinate is
$r=\sqrt{1-(1-t^2)s^2}\in[\sqrt3/2,1]$ when $|s|\leq1/2$.
Substitute $v=rs'$ and retain $|v|\leq1/4$; then $|s'|\leq1/2$
and the Jacobian $1/r$ is at least one.

For $Q=P_1P_2P_1$, write $P_1u=\phi_u(x_{n+2})$.
The two conditional-coordinate formulas give
$Qu=(K^2\phi_u)(x_{n+2})$. Hence \eqref{eq:doeblin} says
$Qu\geq\varepsilon\nu(u)$ for $u\geq0$, where
\[
 \varepsilon=b^2/2\in(0,1),\qquad
                        \nu(u)=2\int_{-1/4}^{1/4}\phi_u(v)\,dv
\]
is a positive normalized functional. The operator
$R=(Q-\varepsilon\nu)/(1-\varepsilon)$ is positive and unital.
For mean-zero $u$, Jensen's inequality and preservation of spherical
measure by $Q$ give
\[
 \|Qu\|_2^2\leq\|Qu-\varepsilon\nu(u)\|_2^2
   \leq(1-\varepsilon)\bigl(\|u\|_2^2-\varepsilon\nu(u^2)\bigr)
   \leq(1-\varepsilon)\|u\|_2^2.
\]
On the other hand,
$\|u-Qu\|_2\leq2\|u-P_1u\|_2+\|u-P_2u\|_2$. Therefore
\[
 \frac{(1-\sqrt{1-\varepsilon})^2}{5}\|u\|_2^2
          \leq\|u-P_1u\|_2^2+\|u-P_2u\|_2^2.
\]
Combining with the fiberwise gaps proves the proposition. In applying
those gaps, sum over the generators before integrating in the
latitude parameter; no measurable choice of a maximizing generator
is required.
\end{proof}

Proposition~\ref{prop:S3gap}, the transfer to $S^2$, and
Proposition~\ref{prop:dimension-gap} give the spectral-gap input
for every $n\geq2$.

\begin{thebibliography}{99}
\bibitem{BG} J. Bourgain and A. Gamburd,
\emph{On the spectral gap for finitely-generated subgroups of
$\mathrm{SU}(2)$}, Invent. Math. \textbf{171} (2008), 83--121.
\url{https://doi.org/10.1007/s00222-007-0072-z}.
\bibitem{CJMST} C. T. Conley, S. Jackson, A. S. Marks, B. Seward,
and R. D. Tucker-Drob, \emph{Hyperfiniteness and Borel combinatorics},
J. Eur. Math. Soc. \textbf{22} (2020), 877--892.
\url{https://doi.org/10.4171/JEMS/935}.
\bibitem{Drinfeld} V. G. Drinfel'd,
\emph{Finitely-additive measures on $S^2$ and $S^3$, invariant with
respect to rotations}, Funct. Anal. Appl. \textbf{18} (1984), 245--246.
\url{https://doi.org/10.1007/BF01086166}.
\bibitem{GJS} A. Gamburd, D. Jakobson, and P. Sarnak,
\emph{Spectra of elements in the group ring of $\mathrm{SU}(2)$},
J. Eur. Math. Soc. \textbf{1} (1999), 51--85.
\url{https://ems.press/journals/jems/articles/99}.
\bibitem{GMP} {\L}. Grabowski, A. M\'ath\'e, and O. Pikhurko,
\emph{Measurable equidecompositions for group actions with an
expansion property}, J. Eur. Math. Soc. \textbf{24} (2022), 4277--4326.
\url{https://doi.org/10.4171/JEMS/1189}.
\bibitem{Kechris} A. S. Kechris, \emph{Classical descriptive set theory},
Graduate Texts in Mathematics 156, Springer, 1995.
\bibitem{Margulis} G. A. Margulis, \emph{Some remarks on invariant means},
Monatsh. Math. \textbf{90} (1980), 233--235.
\url{https://doi.org/10.1007/BF01295368}.
\bibitem{MargulisEucl} G. A. Margulis,
\emph{Finitely-additive invariant measures on Euclidean spaces},
Ergodic Theory Dynam. Systems \textbf{2} (1982), 383--396.
\url{https://doi.org/10.1017/S014338570000167X}.
\bibitem{MT} J. Mycielski and G. Tomkowicz,
\emph{Some theorems related to a problem of Ruziewicz},
Bull. Polish Acad. Sci. Math. \textbf{72} (2024), 1--5.
\url{https://doi.org/10.4064/ba231023-25-12}.
\bibitem{Rosenblatt} J. Rosenblatt,
\emph{Uniqueness of invariant means for measure-preserving transformations},
Trans. Amer. Math. Soc. \textbf{265} (1981), 623--636.
\url{https://doi.org/10.1090/S0002-9947-1981-0610970-7}.
\bibitem{Sullivan} D. Sullivan,
\emph{For $n>3$ there is only one finitely additive rotationally
invariant measure on the $n$-sphere defined on all Lebesgue measurable
subsets}, Bull. Amer. Math. Soc. (N.S.) \textbf{4} (1981), 121--123.
\url{https://doi.org/10.1090/S0273-0979-1981-14880-1}.
\bibitem{Unger} S. Unger, \emph{Equidecomposition and discrepancy,
Part 1}, Winter School 2025 lecture slides.
\url{https://www.winterschool.eu/files/1482-Equidecomposition_and_discrepancy_Part_1.pdf}.
\end{thebibliography}
\end{document}

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