January 14, 2026
We establish a non-ergodic version of the Host–Kra–Ziegler structure theorem for measure-preserving \(\mathbb{Z}^d\)-actions. Our argument reduces the non-ergodic case to the ergodic theorem (for \(d\ge 2\) due to Candela and Szegedy) via a measurable selection procedure. We also establish a non-ergodic vertical nilcharacter version of our main result. The non-ergodic version of the Host–Kra–Ziegler structure theorem is a key input in the companion paper by the second author and Fraczyk classifying point processes (i.e. random subsets) of \(\mathbb{Z}^d\) whose law is invariant under the group \(\mathrm{ASL}_d(\mathbb{Z})\) of affine transformations.
Host and Kra [1] and, independently, Ziegler [2] established the following remarkable structure theorem for ergodic \(\mathbb{Z}\)-actions1. The corresponding conclusion for ergodic \(\mathbb{Z}^d\)-actions follows from a more general theory developed in [4].
Theorem 1. Let \(\mathrm{X}=(X,\nu,T)\) be an ergodic measure-preserving \(\mathbb{Z}^d\)-system and let \(k\ge 1\). Then the \(k\)th Host–Kra factor \(\mathrm Z_k(\mathrm{X})\) of \(\mathrm{X}\) is (measure-theoretically) isomorphic to an inverse limit of \(k\)-step nilsystems.
Informally, the factor \(\mathrm Z_k(\mathrm{X})\) governs the distribution of \((k+1)\) dimensional cubes in \(\mathrm{X}\), and thereby controls the limiting behaviour of \(k\)-linear ergodic averages. Its structural description in geometric terms plays an important role in the understanding of additive patterns such as arithmetic progressions in arithmetic combinatorics. The formal definitions of the Host–Kra factors and of \(k\)-step nilsystems, and the notation appearing in Theorem 2 and Theorem 3 are given in the preliminaries in Section 2.
Motivated by applications, the aim of this note is to remove the ergodicity assumption from Theorem 1. Our main result is the following non-ergodic variant.
Theorem 2. Let \(\mathrm{X}=(X,\nu,T)\) be a not necessarily ergodic measure-preserving \(\mathbb{Z}^d\)-system and let \(k\ge 1\). Then there exists an inverse system of factors \[\mathrm{Y}_1 \twoheadleftarrow \mathrm{Y}_2 \twoheadleftarrow \cdots \twoheadleftarrow \mathrm{X}\] such that:
each \(\mathrm{Y}_n\) is a bundle of \(k\)-step nilsystems over the invariant factor \(\Omega\), and the bundle projection \(\mathrm{Y}_n\to\Omega\) agrees with the projection on the invariant factor;
\(\mathrm Z_k(\mathrm{X})\) is (measure-theoretically) isomorphic to the inverse limit of the \(\mathrm{Y}_n\).
In a companion paper [5], Fraczyk and the second author leverage Theorem 2 to classify random subsets of \(\mathbb{Z}^d\) whose law is invariant under the affine group \(\mathrm{ASL}_d(\mathbb{Z}) = \mathrm{SL}_d(\mathbb{Z}) \ltimes \mathbb{Z}^d\). This classification is driven by considerations in mathematical physics due to Marklof [6], alongside deep structural motivations in dynamics, including exchangeability [7] and discrete analogues of the Howe–Moore phenomenon [8]. The analysis in [5] hinges on a two-step reduction where Theorem 2 plays a pivotal role. Initially, invariance under upper triangular matrices in \(\mathrm{SL}_d(\mathbb{Z})\) reduces the classification to a study of non-conventional ergodic averages, posing a structural problem for \(\mathbb{Z}^d\)-systems of order \(k\). Methods from homogeneous dynamics are then deployed to resolve the resulting questions on bundles of nilsystems under an \(\mathrm{ASL}_d(\mathbb{Z})\)-action. Theorem 2 acts as the crucial bridge connecting these two distinct dynamical environments.
A non-ergodic nilsequence formulation of Theorem 1 for \(\mathbb{Z}\)-actions was established by Chu, Frantzikinakis, and Host, see [9]. This result has served as a key input in several ergodic-theoretic and arithmetic applications; see, e.g., [10]–[13]. A version of Theorem 2 in the case \(k=1\) for \(\mathbb{Z}\)-actions appears in [14]. Closely related non-ergodic Mackey and Furstenberg–Zimmer structure theorems are developed in [15]; see also [16].
We also record a non-ergodic version of the standard fact that ergodic systems of order \(k\) are spanned by vertical nilcharacters.
Theorem 3. Let \(\mathrm X=(X,\nu,T)\) be a measure-preserving \(\mathbb{Z}^d\)-system and let \(k\geq 1\). Then \(L^2(\mathrm Z_k(\mathrm X))\) is the closed linear span of the non-ergodic \(k\)-vertical nilcharacters on \(\mathrm X\).
The case \(k=1\) of Theorem 3 was established by Frantzikinakis and Host, see [17], and has found numerous applications, see, e.g., [11], [18]–[20] for some recent applications.
The authors would like to thank Henrik Kreidler for helpful discussions. We thank Nikos Frantzikinakis for his feedback on earlier versions of the manuscript, in particular for suggesting Theorem 3. A.J. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the Heisenberg Grant – 547294463 and its Excellence Strategy – EXC-2047 – 390685813. S.M. acknowledges support from a Hermann Weyl Instructorship at the Institute for Mathematical Research (FIM) at ETH Zurich.
In this section, we introduce our notation and collect some existing results that will be used in the proof of Theorem 2. We start by defining the Host–Kra factors via the device of Gowers–Host–Kra seminorms as introduced in [1].
Let \(\mathrm{X}=(X,\nu,T)\) be a measure-preserving \(\mathbb{Z}^d\)-system formed on a standard Lebesgue probability space. Since \(X\) is standard Borel, we fix once and for all a Polish topology on \(X\) whose Borel \(\sigma\)-algebra coincides with the given one; this allows us to speak of the weak topology on spaces of probability measures on \(X\) and products of \(X\) with compact metric spaces.
We define measures \(\nu^{[k]}\) on \(X^{[k]}\mathrel{\vcenter{:}}= X^{\{0,1\}^k}\) inductively as follows. Set \(\nu^{[0]}:=\nu\) on \(X^{[0]}:=X\). For \(k\ge 0\), define the diagonal \(\mathbb{Z}^d\)-action \(T^{[k]}\) on \(X^{[k]}\) by \[(T^{[k]})^{t}\bigl((x_\omega)_{\omega\in\{0,1\}^k}\bigr) := \bigl(T^{t}x_\omega\bigr)_{\omega\in\{0,1\}^k}, \qquad t\in\mathbb{Z}^d.\] Let \(\mathcal{I}^{[k]}\) be the \(\sigma\)-algebra of \(T^{[k]}\)-invariant sets in \(X^{[k]}\). Identify \(X^{[k+1]}\) with \(X^{[k]}\times X^{[k]}\) via \[(x_\omega)_{\omega\in\{0,1\}^{k+1}} \longleftrightarrow \Bigl( (x_{\omega 0})_{\omega\in\{0,1\}^k},\;(x_{\omega 1})_{\omega\in\{0,1\}^k} \Bigr).\] Then define \(\nu^{[k+1]}\) to be the relatively independent joining of \(\nu^{[k]}\) with itself over \(\mathcal{I}^{[k]}\), i.e.for all \(f,g\in L^\infty(\nu^{[k]})\), \[\int_{X^{[k+1]}} f(x')\,g(x'')\,d\nu^{[k+1]}(x',x'') := \int_{X^{[k]}} \mathbb{E}(f\mid \mathcal{I}^{[k]})\;\mathbb{E}(g\mid \mathcal{I}^{[k]})\;d\nu^{[k]}.\] For \(f\in L^\infty(\nu)\) and \(k\ge 1\), define \[\left\lVert f\right\rVert_{U^k}^{2^k} := \int_{X^{[k]}} \prod_{\omega\in\{0,1\}^k} \mathcal{C}^{|\omega|} f(x_\omega)\;d\nu^{[k]}((x_\omega)_\omega),\] where \(\mathcal{C}\) denotes complex conjugation and \(|\omega|=\omega_1+\cdots+\omega_k\).
For each \(k\ge 1\) there is a factor \(\mathrm Z_{k-1}=\mathrm Z_{k-1}(\mathrm{X})\) of \(\mathrm{X}\) characterized by the property that, for every \(f\in L^\infty(\nu)\), \[\label{eq:Zk-characterization} \left\lVert f\right\rVert_{U^k}=0 \quad\Longleftrightarrow\quad \mathbb{E}(f\mid \mathrm Z_{k-1})=0.\tag{1}\] We refer to \(\mathrm Z_{k}\) as the \(k\)th Host–Kra factor.
We denote by \[\pi_0:X\to \Omega\] the invariant factor, and write the ergodic decomposition as \[\nu=\int_\Omega \nu_\xi\, d\mu(\xi),\] where \(\xi\mapsto \nu_\xi\) is a measurable probability kernel \(\Omega\to \mathcal{P}(X)\) and \((X,\nu_\xi)\) is ergodic for \(\mu\)-a.e.\(\xi\).
Lemma 1. Let \(\mathrm{X}=(X,\nu,T)\) be a measure-preserving \(\mathbb{Z}^d\)-system and let \(\nu=\int_\Omega \nu_\xi\, d\mu(\xi)\) be its ergodic decomposition. For all \(k\geq 1\) and \(\mu\)-a.e.\(\xi\) one has \(\mathrm Z_k(X,\nu_\xi, T)=(\mathrm Z_k(\mathrm{X}),\nu_\xi, T)\).
Proof. See, e.g., [21]. ◻
We will construct bundles of nilmanifolds over the invariant factor to geometrically represent Host–Kra factors of non-ergodic systems. The following definition formalizes this notion.
Definition 1 (Bundle of nilsystems). Let \((\Omega, \mu)\) be a standard Borel probability space. Choose:
a measurable partition \((\Omega_n) _{n \geq 0}\) of \(\Omega\);
compactly generated nilpotent Lie groups \((N_n)_{n \geq 0}\) with discrete cocompact subgroups (\(=\) lattices) \(\Gamma_n \subset N_n\) for all \(n \geq 0\);
measurable maps \(\phi_n: \Omega_n \rightarrow \mathrm{Hom}(\mathbb{Z}^d,N_n)\) such that for all \(n \geq 0\) and \(\mu\)-almost all \(\xi \in \Omega_n\) the \(\mathbb{Z}^d\)-action given by \(\phi_n(\xi)\) on \((N_n/\Gamma_n, m_{N_n/\Gamma_n})\) is ergodic.
Then the space \[X := \bigsqcup_{n \geq 0} \Omega_n \times N_n/\Gamma_n\] equipped with the probability measure \[\nu:=\sum_{n \geq 0} \mu_{\vert \Omega_n} \otimes m_{N_n/\Gamma_n}\] and the action defined for all \(n \geq 0\), \(t \in \mathbb{Z}^d\) and \((\xi,m) \in \Omega_n \times N_n/\Gamma_n\) by \[t \cdot (\xi,m) = (\xi, \phi_n(\xi)(t)\cdot m)\] is called a bundle of nilsystems (with base space \(\Omega\)) and denoted by \[\bigsqcup_{n \geq 0} \Omega_n \rtimes_{\phi_n} N_n/\Gamma_n .\] We say that \(X\) has step at most \(k\) if \(N_n\) is nilpotent of step at most \(k\) for all \(n \geq 0\). Finally, the obvious projection \(X \rightarrow \Omega\) sends \(\nu\) to \(\mu\) and is called the bundle projection.
Next, we formalize our notion of non-ergodic vertical nilcharacter needed in Theorem 3.
For a nilpotent Lie group \(N\), write \[N_1:=N,\qquad N_{r+1}:=[N,N_r].\] If \(M=N/\Gamma\) is a nilmanifold of step at most \(k\), then \[A_k(M):=N_k/(N_k\cap \Gamma)\] is a compact abelian Lie group. Since \(N_{k+1}=\{e\}\), the group \(N_k\) is central in \(N\), and hence \(A_k(M)\) acts on \(M\) by left translations. If \(N_k=\{e\}\), then \(A_k(M)\) is understood to be the trivial group.
Definition 2 (Vertical nilcharacters over the invariant factor). Let \[\mathrm{Y}=\bigsqcup_{i\geq 0}\Omega_i\rtimes_{\alpha_i} N_i/\Gamma_i\] be a bundle of nilsystems of step at most \(k\) over \((\Omega,\mu)\), in the sense of Definition 1. Put \[M_i:=N_i/\Gamma_i,\qquad A_i:=A_k(M_i)=(N_i)_k/((N_i)_k\cap \Gamma_i).\] A function \(F\in L^2(Y)\) is called a \(k\)-vertical nilcharacter on \(\mathrm Y\) if, for every \(i\geq 0\), there exists a measurable map \(\chi_i:\Omega_i\to \widehat{A_i}\) such that \[F(\xi,a\cdot m)=\chi_i(\xi)(a)\,F(\xi,m)\] for \(\mu|_{\Omega_i}\otimes m_{A_i}\otimes m_{M_i}\)-a.e.\((\xi,a,m)\in \Omega_i\times A_i\times M_i\).
If \(\mathrm X=(X,\nu,T)\) is a measure-preserving system, a function \(f\in L^2(X)\) is called a non-ergodic \(k\)-vertical nilcharacter on \(\mathrm X\) if there exist a factor map \(\pi:\mathrm X\to \mathrm Y\) onto a bundle \(\mathrm{Y}\) of nilsystems of step at most \(k\), and a \(k\)-vertical nilcharacter \(F\in L^2(Y)\), such that \(f=F\circ \pi\) in \(L^2(X)\).
Example 1. Let \[\mathrm{Y}=\bigsqcup_{i\geq 0}\Omega_i\rtimes_{\alpha_i} M_i, \qquad M_i=N_i/\Gamma_i,\] be a bundle of nilsystems of step at most \(1\). Then each \(N_i\) is abelian and \[A_1(M_i)=N_i/(N_i\cap \Gamma_i)=M_i.\] Thus a \(1\)-vertical nilcharacter on \(\mathrm{Y}\) is a function \(F\in L^2(Y)\) such that, for every \(i\geq 0\), there exists a measurable map \(\chi_i:\Omega_i\to \widehat{M_i}\) with \[F(\xi,a\cdot m)=\chi_i(\xi)(a)F(\xi,m)\] for \(\mu|_{\Omega_i}\otimes m_{M_i}\otimes m_{M_i}\)-a.e. \((\xi,a,m)\in\Omega_i\times M_i\times M_i\). Equivalently, for \(\mu\)-a.e. \(\xi\in\Omega_i\), the function \(m\mapsto F(\xi,m)\) is a scalar multiple of a character of the compact abelian Lie group \(M_i\), where the character is allowed to vary measurably with \(\xi\).
In particular, if \[t\cdot(\xi,m)=(\xi,\alpha_i(\xi)(t)\cdot m),\] then \[F(t\cdot(\xi,m)) = \lambda_t(\xi)F(\xi,m), \qquad \lambda_t(\xi):=\chi_i(\xi)\bigl(\alpha_i(\xi)(t)\Gamma_i\bigr).\] Thus, after pulling back to a system \(\mathrm X\) through a factor map \(\pi\colon \mathrm X\to \mathrm Y\), a \(1\)-vertical nilcharacter is a relative eigenfunction over the invariant factor \(\Omega\), with invariant eigenvalue family \((\lambda_t)_{t\in\mathbb{Z}^d}\).
In the case \(d=1\), after the normalization \(\mathbb{E}(|F|^2\mid\Omega)\in\{0,1\}\), this agrees with the notion of non-ergodic eigenfunction used by Frantzikinakis and Host [17].
Remark 4. In the arguments below, some partitions/maps are obtained only \(\mu\)-measurably (e.g.via the Jankov–von Neumann uniformization theorem). Since \(\Omega\) is standard Borel, after modifying on a \(\mu\)-null set one may arrange these to be Borel. We will freely work modulo null sets. In particular, we will (after modifying on a \(\mu\)-null set) regard the kernel \(\xi\mapsto \nu_\xi\in\mathcal{P}(X)\) as Borel.
To perform our measurable selection argument, we need the following countability result for nilmanifolds.
Theorem 5. Up to isomorphism of nilmanifolds, there are only countably many pairs \((N,\Gamma)\) where \(N\) is a compactly generated nilpotent Lie group and \(\Gamma\) is a discrete cocompact subgroup.
Proof. See, e.g., [21]. ◻
Our measurable selection argument relies on the Jankov–von Neumann uniformization theorem:
Theorem 6. Let \(A\subseteq U\times V\) be a Borel subset of a product of standard Borel spaces. If every section \(A_u:=\{v:(u,v)\in A\}\) is nonempty for \(u\) in a measurable set \(U_0\subseteq U\), then there exists a universally measurable map \(s:U_0\to V\) with \((u,s(u))\in A\) for all \(u\in U_0\).
Proof. See, e.g., [22]. ◻
Finally, we need the following disintegration result of Kallenberg.
Lemma 2. Let \(V,W\) be standard Borel spaces. There exists a Borel map \[\Phi:\;V\times \mathcal{P}(V\times W)\times \mathcal{P}(V)\;\longrightarrow\;\mathcal{P}(W)\] such that whenever \(\lambda\in\mathcal{P}(V\times W)\) and \(\nu\in\mathcal{P}(V)\) satisfy \(\lambda(\cdot\times W)\ll \nu\), one has the disintegration identity \[\lambda(A\times B)=\int_A \Phi(s,\lambda,\nu)(B)\, d\nu(s) \qquad \text{for all Borel }A\subseteq V,\;B\subseteq W.\] In particular, for fixed \(\nu\), the map \((s,\lambda)\mapsto \Phi(s,\lambda,\nu)\) is jointly Borel.
Proof. See [23]. ◻
It suffices to prove Theorem 2 under the additional assumption that \[\mathrm{X}=\mathrm Z_k(\mathrm{X}).\] Indeed, in the general case one applies the result below to the factor \(\mathrm Z_k(\mathrm{X})\) and then pulls back the resulting inverse system to \(\mathrm{X}\). The invariant factor of \(\mathrm Z_k(\mathrm{X})\) is the same as the invariant factor of \(\mathrm{X}\), since the invariant factor is contained in every Host–Kra factor.
We fix the following setting throughout the proof of this reduced case.
\(k\geq 1\).
A measure-preserving \(\mathbb{Z}^d\)-system \(\mathrm{X}=(X,\nu,T)\) such that \(\mathrm{X} =\mathrm Z_k(\mathrm{X})\).
Let \(\nu=\int_\Omega \nu_\xi\, d\mu(\xi)\) be the ergodic decomposition of \(\mathrm{X}\).
A countable set \(\mathcal{D}=\{f_r:r\in\mathbb{N}\}\subset L^\infty(\nu)\) dense in \(L^2(\nu)\).
An enumeration of representatives of \(k\)-step nilmanifolds (using Theorem 5): \[(N_j,\Gamma_j)_{j\in\mathbb{N}}, \qquad M_j:=N_j/\Gamma_j.\]
For each \(j\), a compatible metric \(d_j\) on \(M_j\) and a countable set \(\mathcal{U}_j=\{u_{j,q}\}_{q\in\mathbb{N}}\subset C(M_j)\) dense in \(L^2(m_{M_j})\).
For each \(j\), let \(\mathcal{P}(X\times M_j)\) be the Polish space of Borel probability measures on \(X\times M_j\) with the weak topology, and let \[\mathsf Q:=\bigsqcup_{j\in\mathbb{N}}\mathcal{P}(X\times M_j)\] be the disjoint union standard Borel space of all these spaces of measures.
For each \(j\), let \[\mathsf P_j:=\mathrm{Hom}(\mathbb{Z}^d,N_j),\] be identified with the closed subset of \(N_j^d\) consisting of commuting \(d\)-tuples (hence \(\mathsf P_j\) is Polish), and let \[\mathsf P:=\bigsqcup_{j\in\mathbb{N}}\mathsf P_j\] be the disjoint union standard Borel space of all such actions.
Lemma 3. For \(\mu\)-a.e.\(\xi\) and for every \(n\in\mathbb{N}\) there exist \(j\in\mathbb{N}\), an action \(\alpha\in\mathsf P_j\), and a \(k\)-step nilfactor map \[\phi_{\xi,n}:(X,\nu_\xi)\to (M_j,m_{M_j})\] intertwining \(T\) and \(\alpha\), such that the action \(\alpha\) on \((M_j,m_{M_j})\) is ergodic and such that for all \(1\le r\le n\), \[\big\| f_r - \mathbb{E}_{\nu_\xi}(f_r \mid \phi_{\xi,n}^{-1}\mathcal{B}(M_j))\big\|_{L^2(\nu_\xi)} < 2^{-(n+2)}.\]
Proof. Fix \(\xi\) such that \((X,\nu_\xi,T)\) is ergodic. By Lemma 1 and Theorem 1, \((X,\nu_\xi,T)\) is an inverse limit of \(k\)-step nilsystems. Hence for the finite set \(\{f_1,\dots,f_n\}\) and tolerance \(2^{-(n+2)}\), there is a \(k\)-step nilfactor \(\phi_{\xi,n}\) such that each \(f_r\) is within \(2^{-(n+2)}\) in \(L^2(\nu_\xi)\) of its conditional expectation onto that factor. By Theorem 5 we may take the nilfactor isomorphic to some \(M_j\) from our countable list, with action given by some \(\alpha\in\mathsf P_j\). Since this nilsystem is a factor of the ergodic system \((X,\nu_\xi,T)\), the action \(\alpha\) on \((M_j,m_{M_j})\) is ergodic. ◻
If \(\phi:(X,\nu_\xi)\to (M_j,m_{M_j})\) is a factor map, define its graph joining \[\lambda:=(\mathrm{id}_X,\phi)_*\nu_\xi=\int_X \delta_x\times \delta_{\phi(x)} \, d\nu_\xi \in \mathcal{P}(X\times M_j).\] Then \((\mathrm{pr}_X)_*\lambda=\nu_\xi\) and \((\mathrm{pr}_{M_j})_*\lambda=m_{M_j}\).
Fix \(j\in\mathbb{N}\). For \((\xi,\lambda,x)\) with \((\mathrm{pr}_X)_*\lambda=\nu_\xi\), define the conditional measure \[\lambda_x := \Phi(x,\lambda,\nu_\xi)\in \mathcal{P}(M_j),\] where \(\Phi\) is given by Lemma 2. By Remark 4, the map \(\xi\mapsto \nu_\xi\) is Borel, and hence \[(\xi,\lambda,x)\longmapsto \lambda_x\] is Borel on its natural domain \(\{(\xi,\lambda,x):(\mathrm{pr}_X)_*\lambda=\nu_\xi\}\).
Lemma 4. Fix \(j\in\mathbb{N}\). Consider the Dirac embedding \[\delta:M_j\to \mathcal{P}(M_j),\qquad m\mapsto \delta_m.\] Then \(\delta\) is continuous and injective, its image \(\delta(M_j)\subseteq \mathcal{P}(M_j)\) is Borel, and the inverse \(\delta^{-1}:\delta(M_j)\to M_j\) is Borel.
Define the Borel set \[\mathcal{G}_j:=\{(\xi,\lambda,x): (\mathrm{pr}_X)_*\lambda=\nu_\xi \;\text{and}\; \lambda_x\in \delta(M_j)\}.\] On \(\mathcal{G}_j\) define \[\Psi_j(\xi,\lambda,x):=\delta^{-1}(\lambda_x)\in M_j.\] Then \(\Psi_j:\mathcal{G}_j\to M_j\) is Borel.
Moreover, for \((\xi,\lambda)\) with \((\mathrm{pr}_X)_*\lambda=\nu_\xi\) and \((\mathrm{pr}_{M_j})_*\lambda=m_{M_j}\), the following are equivalent:
\(\lambda\) is supported on the graph of a measurable map \(X\to M_j\);
\(\lambda_x\) is a Dirac mass for \(\nu_\xi\)-a.e.\(x\);
one has \[\int_X \left(\int_{M_j}\int_{M_j} d_j(m,m')\,d\lambda_x(m)\,d\lambda_x(m')\right)\,d\nu_\xi(x)=0.\]
In particular, the set of pairs \((\xi,\lambda)\) with \((\mathrm{pr}_X)_*\lambda=\nu_\xi\), \((\mathrm{pr}_{M_j})_*\lambda=m_{M_j}\) and \(\lambda\) a graph joining is Borel in \(\Omega\times \mathcal{P}(X\times M_j)\).
Finally, whenever \((\xi,\lambda)\) is a graph joining, the map \(x\mapsto \Psi_j(\xi,\lambda,x)\) is defined \(\nu_\xi\)-a.e.and satisfies \[(\mathrm{id}_X,\Psi_j(\xi,\lambda,\cdot))_*\nu_\xi=\lambda.\] We denote this a.e.defined map by \(\phi_\lambda\).
Proof. The first paragraph follows from the Lusin–Souslin theorem (e.g.[22]) applied to the continuous injective map \(\delta\).
Borelness of \(\mathcal{G}_j\) and \(\Psi_j\) follows because \((\xi,\lambda,x)\mapsto \lambda_x\) is Borel on \(\{(\mathrm{pr}_X)_*\lambda=\nu_\xi\}\), the set \(\delta(M_j)\) is Borel in \(\mathcal{P}(M_j)\), and \(\delta^{-1}\) is Borel on its domain.
For the equivalences: (i)\(\Leftrightarrow\)(ii) is the definition of a graph joining in terms of conditional measures. (ii)\(\Leftrightarrow\)(iii) uses the metric characterization of Dirac masses: for any probability measure \(\eta\) on a metric space, \(\eta\) is a Dirac mass if and only if \(\int\!\!\int d(m,m')\,d\eta(m)\,d\eta(m')=0\). Apply this to \(\eta=\lambda_x\) for \(\nu_\xi\)-a.e.\(x\) and integrate over \(x\).
For the final statement, if \(\lambda_x=\delta_{\psi(x)}\) \(\nu_\xi\)-a.e., then on the \(\nu_\xi\)-conull set where \(\lambda_x\in\delta(M_j)\) we have \(\psi(x)=\delta^{-1}(\lambda_x)=\Psi_j(\xi,\lambda,x)\), hence \[(\mathrm{id}_X,\Psi_j(\xi,\lambda,\cdot))_*\nu_\xi=\lambda.\] ◻
Fix \(n\in\mathbb{N}\) and define a relation \(R_n\subseteq \Omega\times \mathsf P\times \mathsf Q\) by \[(\xi;\;(j,\alpha),\;\lambda)\in R_n\] if and only if all of the following hold:
\(\lambda\in\mathcal{P}(X\times M_j)\) has marginals \((\mathrm{pr}_X)_*\lambda=\nu_\xi\), \((\mathrm{pr}_{M_j})_*\lambda=m_{M_j}\), and \(\lambda\) is a graph joining.
For every \(t\in\mathbb{Z}^d\), \[(T^t\times \alpha(t))_*\lambda=\lambda.\]
For each \(1\le r\le n\) there exists \(q\in \mathbb{N}\) such that \[\int_{X\times M_j} \bigl|f_r(x)-u_{j,q}(m)\bigr|^2\,d\lambda(x,m) < 2^{-2n}.\]
The action \(\alpha\) on \((M_j,m_{M_j})\) is ergodic.
Lemma 5. For each \(n\in\mathbb{N}\), the set \(R_n\) is a Borel subset of \(\Omega \times \mathsf P \times \mathsf Q\).
Proof. Borelness of (R2). Fix \(t\in\mathbb{Z}^d\). For each \(j\), the evaluation map \[\mathsf P_j\to N_j,\qquad \alpha\mapsto \alpha(t)\] is continuous (indeed, \(\mathsf P_j\) is a closed subset of \(N_j^d\) and \(\alpha(t)\) is a word in the \(d\) generators). The action map \(N_j\times M_j\to M_j\) is continuous, hence the map \[\Xi_{t,j}: \mathsf P_j\times (X\times M_j)\to X\times M_j,\qquad (\alpha,(x,m))\mapsto (T^t x,\;\alpha(t)\cdot m)\] is Borel. Therefore the induced pushforward map \[(\alpha,\lambda)\;\longmapsto\;(\Xi_{t,j}(\alpha,\cdot))_*\lambda = (T^t\times \alpha(t))_*\lambda\] from \(\mathsf P_j\times \mathcal{P}(X\times M_j)\) to \(\mathcal{P}(X\times M_j)\) is Borel. Consequently, the set \[A_{t}:=\bigl\{(\xi,(j,\alpha),\lambda):\;(T^t\times \alpha(t))_*\lambda=\lambda\bigr\}\] is Borel, since it is the preimage of the diagonal under \[(\xi,(j,\alpha),\lambda)\mapsto\bigl((T^t\times \alpha(t))_*\lambda,\lambda\bigr).\] Since \(\mathbb{Z}^d\) is countable, \(\bigcap_{t\in\mathbb{Z}^d}A_t\) is Borel, proving that (R2) is Borel.
Borelness of (R3). Fix \(1\le r\le n\) and \(q\in\mathbb{N}\). On the component indexed by \(j\) define \[I_{r,q}(\xi,(j,\alpha),\lambda) :=\int_{X\times M_j} \bigl|f_r(x)-u_{j,q}(m)\bigr|^2\,d\lambda(x,m).\] Because \((x,m)\mapsto |f_r(x)-u_{j,q}(m)|^2\) is bounded Borel and \(\lambda\mapsto \int \varphi\,d\lambda\) is Borel for bounded Borel \(\varphi\), the map \(I_{r,q}\) is Borel. Hence \[B_{r,q}:=\bigl\{(\xi,(j,\alpha),\lambda):\;I_{r,q}(\xi,(j,\alpha),\lambda)<2^{-2n}\bigr\}\] is Borel. Now the clause (R3) for a fixed \(r\) is \(\exists q\in\mathbb{N}\) with \(I_{r,q}<2^{-2n}\), i.e. \[B_r:=\bigcup_{q\in\mathbb{N}}B_{r,q},\] a countable union of Borel sets, hence Borel. Intersecting over \(r=1,\dots,n\) shows (R3) is Borel.
Borelness of (R4). Let \[F_N:=\{-N,\dots,N\}^d\subseteq \mathbb{Z}^d.\] For \(u\in C(M_j)\) and \(\alpha\in\mathsf P_j\), define \[A_N^\alpha u(m):=\frac{1}{|F_N|}\sum_{t\in F_N}u(\alpha(t)\cdot m).\] By the mean ergodic theorem, the action \(\alpha\) is ergodic if and only if, for every \(q\in\mathbb{N}\), \[\lim_{N\to\infty} \int_{M_j}\left|A_N^\alpha u_{j,q}(m)-\int_{M_j}u_{j,q}\,dm_{M_j}\right|^2\,dm_{M_j}(m)=0.\] This equivalence uses that \(\mathcal{U}_j\) is dense in \(L^2(m_{M_j})\). For fixed \(q\) and \(N\), the displayed integral is a Borel, indeed continuous, function of \(\alpha\). Therefore the set of ergodic \(\alpha\in\mathsf P_j\) is Borel, being \[\bigcap_{q\in\mathbb{N}}\bigcap_{\ell\in\mathbb{N}} \bigcup_{N_0\in\mathbb{N}}\bigcap_{N\geq N_0} \left\{ \alpha: \int_{M_j}\left|A_N^\alpha u_{j,q}-\int_{M_j}u_{j,q}\,dm_{M_j}\right|^2\,dm_{M_j}<\frac{1}{\ell} \right\}.\] Hence (R4) is Borel.
Borelness of (R1). Fix \(j\). The marginal maps \[\lambda\mapsto (\mathrm{pr}_X)_*\lambda \in \mathcal{P}(X), \qquad \lambda\mapsto (\mathrm{pr}_{M_j})_*\lambda \in \mathcal{P}(M_j)\] are continuous (hence Borel) for weak topologies. Since \(\xi\mapsto \nu_\xi\in\mathcal{P}(X)\) is Borel (Remark 4), the set of \((\xi,\lambda)\) with \((\mathrm{pr}_X)_*\lambda=\nu_\xi\) is Borel, and the set with \((\mathrm{pr}_{M_j})_*\lambda=m_{M_j}\) is also Borel. Finally, the graph-joining clause is Borel by Lemma 4. Hence (R1) is Borel. ◻
Lemma 6. \((R_n)_\xi\) is nonempty for \(\mu\)-a.e.\(\xi\).
Proof. Fix \(\xi\) such that \((X,\nu_\xi, T)\) is ergodic. By Lemma 3, there is a \(k\)-step nilfactor \(\phi:(X,\nu_\xi)\to(M_j,m_{M_j})\) with action \(\alpha\in\mathsf P_j\) such that each \(f_r\) (\(r\le n\)) is within \(2^{-(n+2)}\) in \(L^2(\nu_\xi)\) of \(\mathbb{E}_{\nu_\xi}(f_r\mid \phi^{-1}\mathcal{B}(M_j))\), and such that \(\alpha\) is ergodic on \((M_j,m_{M_j})\).
Let \(\lambda=(\mathrm{id},\phi)_*\nu_\xi\). Then (R1), (R2), and (R4) hold.
For (R3): for each \(r\le n\), since \(\mathbb{E}_{\nu_\xi}(f_r\mid \phi^{-1}\mathcal{B}(M_j))\) is of the form \(g_r\circ\phi\) with \(g_r\in L^2(M_j)\), we have \[\left\lVert f_r-g_r\circ\phi\right\rVert_{L^2(\nu_\xi)}<2^{-(n+2)}.\] Approximate \(g_r\) in \(L^2(M_j)\) by some \(u_{j,q}\) from the dense set \(\mathcal{U}_j\) so that \[\left\lVert g_r-u_{j,q}\right\rVert_{L^2(M_j)}<2^{-(n+2)}.\] Then \[\left\lVert f_r-u_{j,q}\circ\phi\right\rVert_{L^2(\nu_\xi)} \le \left\lVert f_r-g_r\circ\phi\right\rVert_{L^2(\nu_\xi)}+\left\lVert g_r-u_{j,q}\right\rVert_{L^2(M_j)} <2^{-(n+1)},\] and hence \[\int_{X\times M_j} \bigl|f_r(x)-u_{j,q}(m)\bigr|^2\,d\lambda(x,m) = \left\lVert f_r-u_{j,q}\circ\phi\right\rVert_{L^2(\nu_\xi)}^2 <2^{-2n}.\] Thus (R3) holds for suitable choices of \(q\) depending on \(r\). Hence \((\xi;(j,\alpha),\lambda)\in R_n\). ◻
We can apply Theorem 6 to \(R_n\). After discarding a \(\mu\)-null set, obtain a \(\mu\)-measurable selector \[s_n(\xi)=\bigl((j_n(\xi),\alpha_n(\xi)),\lambda_{n,\xi}\bigr)\] with \((\xi;s_n(\xi))\in R_n\) for \(\mu\)-a.e.\(\xi\).
Let \(\Omega_{n,j}:=\{\xi:j_n(\xi)=j\}\). Define \[Y_n:=\bigsqcup_{j\in\mathbb{N}}(\Omega_{n,j}\times M_j), \qquad \lambda_n:=\sum_{j\in\mathbb{N}}\mu|_{\Omega_{n,j}}\otimes m_{M_j},\] and define the \(\mathbb{Z}^d\)-action on \(Y_n\) by \[t\cdot(\xi,m):=(\xi,\alpha_n(\xi)(t)\cdot m).\] By (R4), this is a bundle of \(k\)-step nilsystems over \(\Omega\) in the sense of Definition 1.
Define the factor map \(\pi_n:X\to Y_n\) by \[\pi_n(x):=\bigl(\pi_0(x),\,\phi_{n,\pi_0(x)}(x)\bigr),\] where \(\phi_{n,\xi}:=\phi_{\lambda_{n,\xi}}\) is extracted from \(\lambda_{n,\xi}\) via Lemma 4 applied on the component \(M_{j_n(\xi)}\).
To make \(\pi_n\) everywhere-defined, fix for each \(j\) a basepoint \(m_{j,0}\in M_j\) and define \(\phi_{n,\xi}(x):=m_{j,0}\) on the \(\nu_\xi\)-null set where \(\phi_{\lambda_{n,\xi}}\) is not defined; this does not affect any of the measure-theoretic identities below.
For each \(n\), the map \(\pi_n\) is measurable, satisfies \[\pi_n(T^t x)=t\cdot \pi_n(x)\quad\text{for }\nu\text{-a.e.\;}x,\;\forall t\in\mathbb{Z}^d,\] and \((\pi_n)_*\nu=\lambda_n\). In particular \((Y_n,\lambda_n)\) is a factor of \((X,\nu)\) and the bundle projection \(Y_n\to\Omega\) equals \(\pi_0\).
Indeed, \(\xi\mapsto \lambda_{n,\xi}\) is \(\mu\)-measurable by construction, and Lemma 4 provides a jointly measurable realization \((\xi,\lambda,x)\mapsto \phi_\lambda(x)\) on graph joinings. Hence \(x\mapsto \pi_n(x)\) is measurable.
For \(\mu\)-a.e.\(\xi\), (R2) says \((T^t\times\alpha_n(\xi)(t))_*\lambda_{n,\xi}=\lambda_{n,\xi}\) for all \(t\). Since \(\lambda_{n,\xi}\) is a graph joining, this is equivalent to \(\phi_{n,\xi}(T^t x)=\alpha_n(\xi)(t)\phi_{n,\xi}(x)\) for \(\nu_\xi\)-a.e.\(x\). Integrating over \(\xi\) yields the stated equivariance \(\nu\)-a.e.on \(X\).
Disintegrate \[(\pi_n)_*\nu=\int_\Omega (\pi_n)_*\nu_\xi\,d\mu(\xi).\] For \(\mu\)-a.e.\(\xi\), (R1) gives \((\mathrm{pr}_{M_{j_n(\xi)}})_*\lambda_{n,\xi}=m_{M_{j_n(\xi)}}\) and \((\mathrm{pr}_X)_*\lambda_{n,\xi}=\nu_\xi\), so \((\pi_n)_*\nu_\xi=\delta_\xi\otimes m_{M_{j_n(\xi)}}\). Integrating gives \(\lambda_n\). Finally, the bundle projection composed with \(\pi_n\) is \(\pi_0\) by definition.
The factors \(Y_n\) need not be increasing in \(n\), so we take finite joins.
Let \(\mathcal{A}_n:=\sigma(\pi_1,\dots,\pi_n)\), and let \(Z_n\) be the corresponding factor of \(X\). Equivalently, \(Z_n\) is the factor induced by \[\Pi_n:X\to \prod_{i=1}^n Y_i,\qquad \Pi_n(x):=(\pi_1(x),\dots,\pi_n(x)).\]
Lemma 7. For each \(n\), the factor \(Z_n\) is a bundle of \(k\)-step nilsystems over \(\Omega\) with bundle projection induced by \(\pi_0\).
Proof. For \(\ell=1,\dots,n\), write \[Y_\ell=\bigsqcup_{i\geq 0}\Omega_i^{(\ell)}\rtimes_{\alpha_i^{(\ell)}} M_i,\] where \(M_i=N_i/\Gamma_i\) is one of the fixed representatives. For a tuple \(\mathbf{i}=(i_1,\dots,i_n)\), put \[\Omega_{\mathbf{i}}:=\bigcap_{\ell=1}^n \Omega_{i_\ell}^{(\ell)},\qquad M_{\mathbf{i}}:=M_{i_1}\times\cdots\times M_{i_n},\] and \[N_{\mathbf{i}}:=N_{i_1}\times\cdots\times N_{i_n}, \qquad \Gamma_{\mathbf{i}}:=\Gamma_{i_1}\times\cdots\times \Gamma_{i_n}.\] For \(\xi\in\Omega_{\mathbf{i}}\), let \[\eta_\xi:=(\Pi_n)_*\nu_\xi\in\mathcal{P}(M_{\mathbf{i}})\] after the evident identification of the fiber over \(\xi\) with \(M_{\mathbf{i}}\). The map \(\xi\mapsto\eta_\xi\) is Borel. Let \[\beta_\xi(t):= \bigl(\alpha_{i_1}^{(1)}(\xi)(t),\dots,\alpha_{i_n}^{(n)}(\xi)(t)\bigr) \in N_{\mathbf{i}}.\] Then \(\beta_\xi\in\mathrm{Hom}(\mathbb{Z}^d,N_{\mathbf{i}})\) depends measurably on \(\xi\), and \(\eta_\xi\) is invariant and ergodic under the nilrotation action given by \(\beta_\xi\). Ergodicity follows because \(\eta_\xi\) is the image of the ergodic system \((X,\nu_\xi,T)\).
We use the standard theorem that an ergodic joining of finitely many ergodic nilsystems is Haar measure on a closed subnilmanifold of the product nilmanifold, and that the product nilrotation restricts to an ergodic nilrotation on that subnilmanifold [21]. Hence, for \(\mu\)-a.e.\(\xi\in\Omega_{\mathbf{i}}\), the measure \(\eta_\xi\) is Haar measure on a subnilmanifold of \(M_{\mathbf{i}}\) of step at most \(k\).
We also use the following standard measurable trivialization consequence of the countability of isomorphism classes of nilmanifolds (Theorem 5) and the Jankov–von Neumann uniformization theorem (Theorem 6).
Claim 7. Let \(M=N/\Gamma\) be a nilmanifold of step at most \(k\). Let \(\mathcal{S}(M)\) be the standard Borel set of pairs \((\eta,\beta)\), where \(\eta\in\mathcal{P}(M)\) is Haar measure on a subnilmanifold of \(M\), \(\beta\in\mathrm{Hom}(\mathbb{Z}^d,N)\) preserves \(\eta\), and the induced action on \((\mathrm{supp}\eta,\eta)\) is ergodic. Then there exist a countable family of \(k\)-step nilmanifolds \[W_q=H_q/\Lambda_q,\qquad q\geq 0,\] a Borel partition \(\mathcal{S}(M)=\bigsqcup_{q\geq 0}\mathcal{S}_q\), and, for each \(q\), Borel assignments \[(\eta,\beta)\in\mathcal{S}_q \longmapsto \theta_{\eta,\beta}:W_q\to \mathrm{supp}\eta\] and \[(\eta,\beta)\in\mathcal{S}_q \longmapsto \widetilde{\beta}_{\eta,\beta}\in\mathrm{Hom}(\mathbb{Z}^d,H_q)\] such that \(\theta_{\eta,\beta}\) is a measure-preserving nilmanifold isomorphism from \((W_q,m_{W_q})\) to \((\mathrm{supp}\eta,\eta)\) and \[\theta_{\eta,\beta}\bigl(\widetilde{\beta}_{\eta,\beta}(t)\cdot w\bigr) = \beta(t)\cdot \theta_{\eta,\beta}(w)\] for all \(t\in\mathbb{Z}^d\) and \(m_{W_q}\)-a.e.\(w\in W_q\).
Applying Claim 7 to \(M_{\mathbf{i}}\) and to the Borel map \[\xi\mapsto(\eta_\xi,\beta_\xi),\] we obtain a countable measurable partition of \(\Omega_{\mathbf{i}}\) into sets \(\Omega_{\mathbf{i},q}\) and, on each \(\Omega_{\mathbf{i},q}\), a fixed nilmanifold \(W_q=H_q/\Lambda_q\) together with measurable homomorphisms \[\xi\mapsto \widetilde{\beta}_\xi\in\mathrm{Hom}(\mathbb{Z}^d,H_q)\] such that the fiber of \(Z_n\) over \(\xi\) is identified with \((W_q,m_{W_q})\) and the action becomes \[t\cdot(\xi,w)=(\xi,\widetilde{\beta}_\xi(t)\cdot w).\] The action is ergodic for \(\mu\)-a.e.\(\xi\) by construction. Since the collection of tuples \(\mathbf{i}\) and indices \(q\) is countable, this gives a countable measurable partition of \(\Omega\) over which \(Z_n\) is represented as a bundle of \(k\)-step nilsystems. The bundle projection is the projection to \(\Omega\), which agrees with \(\pi_0\) because each \(\pi_\ell\) lies over \(\pi_0\). ◻
Lemma 8. For every \(r\in\mathbb{N}\), \[\lim_{n\to\infty}\big\| f_r-\mathbb{E}_\nu(f_r\mid \mathcal{A}_n)\big\|_{L^2(\nu)}=0.\]
Proof. Fix \(r\) and take \(n\ge r\). Since \(\mathcal{A}_n=\sigma(\pi_1,\dots,\pi_n)\) contains \(\sigma(\pi_n)\), \[\|f_r-\mathbb{E}_\nu(f_r\mid \mathcal{A}_n)\|_{2}\le \|f_r-\mathbb{E}_\nu(f_r\mid \sigma(\pi_n))\|_{2}.\] Disintegrating over \(\Omega\) and using that \(\pi_n\) restricts fiberwise to \(\phi_{n,\xi}\) gives \[\|f_r-\mathbb{E}_\nu(f_r\mid \sigma(\pi_n))\|_2^2 =\int_\Omega \|f_r-\mathbb{E}_{\nu_\xi}(f_r\mid \phi_{n,\xi}^{-1}\mathcal{B}(M_{j_n(\xi)}))\|_{L^2(\nu_\xi)}^2\,d\mu(\xi).\] But \((\xi;s_n(\xi))\in R_n\) implies that on \((X,\nu_\xi)\) the function \(f_r\) is within \(2^{-n}\) in \(L^2\) of the factor \(\phi_{n,\xi}^{-1}\mathcal{B}(M_{j_n(\xi)})\), since (R3) gives a factor-measurable approximation and conditional expectation is the best \(L^2\)-approximation onto the factor. Thus the integrand is at most \(2^{-2n}\) for \(\mu\)-a.e.\(\xi\), and the claim follows. ◻
Lemma 9. Let \(\mathcal{A}_\infty:=\bigvee_{n\ge 1}\mathcal{A}_n\). Then \(\mathcal{A}_\infty\) coincides with the full \(\sigma\)-algebra of \(X\) modulo \(\nu\)-null sets.
Proof. Let \(g\in L^2(X,\nu)\) be orthogonal to \(L^2(\mathcal{A}_\infty)\). Then \(\mathbb{E}(g\mid \mathcal{A}_n)=0\) for all \(n\).
Fix \(\varepsilon>0\) and choose \(f_r\in\mathcal{D}\) such that \(\|g-f_r\|_2<\varepsilon\). By Lemma 8, choose \(n\ge r\) such that \[\|f_r-\mathbb{E}(f_r\mid \mathcal{A}_n)\|_2<\varepsilon.\] Now use the orthogonality of \(g\) to \(L^2(\mathcal{A}_n)\): since \(\mathbb{E}(f_r\mid \mathcal{A}_n)\in L^2(\mathcal{A}_n)\), \[\langle g,\, \mathbb{E}(f_r\mid \mathcal{A}_n)\rangle = \langle \mathbb{E}(g\mid \mathcal{A}_n),\, f_r\rangle =0.\] Therefore, \[\begin{align} \|g\|_2^2 &=\langle g,g\rangle\\ &=\langle g,\, g-f_r\rangle + \langle g,\, f_r-\mathbb{E}(f_r\mid \mathcal{A}_n)\rangle + \langle g,\, \mathbb{E}(f_r\mid \mathcal{A}_n)\rangle\\ &=\langle g,\, g-f_r\rangle + \langle g,\, f_r-\mathbb{E}(f_r\mid \mathcal{A}_n)\rangle. \end{align}\] By Cauchy–Schwarz, \[\|g\|_2^2 \le \|g\|_2\,\|g-f_r\|_2 + \|g\|_2\,\|f_r-\mathbb{E}(f_r\mid \mathcal{A}_n)\|_2 < 2\varepsilon\,\|g\|_2.\] If \(\|g\|_2>0\), divide by \(\|g\|_2\) to get \(\|g\|_2<2\varepsilon\); since \(\varepsilon>0\) was arbitrary, \(\|g\|_2=0\). Hence \(g=0\), so \(L^2(\mathcal{A}_\infty)=L^2(X)\) and \(\mathcal{A}_\infty\) is the full \(\sigma\)-algebra modulo null sets. ◻
Conclusion of Theorem 2. The factors \(Z_n\) form an increasing sequence of factors, and the natural maps \(Z_{n+1}\twoheadrightarrow Z_n\) give an inverse system \[Z_1 \twoheadleftarrow Z_2 \twoheadleftarrow \cdots \twoheadleftarrow X.\] By Lemma 7, each \(Z_n\) is a bundle of \(k\)-step nilsystems over \(\Omega\). Lemma 9 shows that \(\bigvee_n \sigma(Z_n)\) is the full \(\sigma\)-algebra of \(X\) modulo null sets, hence \(X\) is measurably isomorphic to the inverse limit \(\varprojlim_n Z_n\). This proves the reduced case \(\mathrm X=\mathrm Z_k(\mathrm X)\), and therefore, by the initial reduction, proves Theorem 2.
Without loss of generality, we can assume that \(\mathrm X=\mathrm Z_k(\mathrm X)\). By Theorem 2, there is an inverse system of factors \[\mathrm{Y}_1 \twoheadleftarrow \mathrm{Y}_2 \twoheadleftarrow \cdots \twoheadleftarrow \mathrm{X}\] such that each \(\mathrm{Y}_n\) is a bundle of nilsystems of step at most \(k\) over the invariant factor \(\Omega\), and \(\mathrm{X}\cong \varprojlim_n \mathrm{Y}_n\) measure-theoretically. Hence \[L^2(\nu)=\overline{\bigcup_{n\geq 1}L^2(\mathrm{Y}_n)},\] where each \(L^2(\mathrm{Y}_n)\) is identified with its pullback to \(L^2(\nu)\). It is therefore enough to prove that, for every bundle \(\mathrm Y\) of nilsystems of step at most \(k\), the space \(L^2(\mathrm{Y})\) is the closed linear span of the \(k\)-vertical nilcharacters on \(\mathrm{Y}\).
Fix such a bundle \[\mathrm{Y}=\bigsqcup_{i\geq 0}\Omega_i\rtimes_{\alpha_i} M_i, \qquad M_i=N_i/\Gamma_i.\] Since the partition \((\Omega_i)_{i\geq 0}\) is countable, it suffices to work on a single component \(\Omega_i\times M_i\). We suppress the index \(i\) and write \[M=N/\Gamma,\qquad A:=N_k/(N_k\cap\Gamma).\]
For each \(\chi\in \widehat A\), define \[\mathcal{H}_\chi := \Bigl\{ F\in L^2(\Omega_i\times M): F(\xi,a\cdot m)=\chi(a)F(\xi,m) \text{ for } \mu|_{\Omega_i}\otimes m_A\otimes m_M\text{-a.e. }(\xi,a,m) \Bigr\}.\] Every element of \(\mathcal{H}_\chi\), extended by \(0\) outside \(\Omega_i\times M\), is a \(k\)-vertical nilcharacter on \(\mathrm Y\). Thus it remains to show that \[L^2(\Omega_i\times M) = \overline{\operatorname{span}}\bigcup_{\chi\in\widehat A}\mathcal{H}_\chi.\]
Let \(U_a\) denote the unitary action of \(A\) on \(L^2(\Omega_i\times M)\) given by \[(U_aF)(\xi,m):=F(\xi,a\cdot m).\] For \(\chi\in\widehat A\), define \[P_\chi F := \int_A U_aF\,\overline{\chi(a)}\,dm_A(a),\] where the integral is taken in \(L^2(\Omega_i\times M)\). Then \(P_\chi\) is the orthogonal projection onto \(\mathcal{H}_\chi\). Indeed, for \(b\in A\), \[\begin{align} (P_\chi F)(\xi,b\cdot m) &= \int_A F(\xi,a b\cdot m)\,\overline{\chi(a)}\,dm_A(a)\\ &= \chi(b)\int_A F(\xi,c\cdot m)\,\overline{\chi(c)}\,dm_A(c)\\ &= \chi(b)(P_\chi F)(\xi,m), \end{align}\] and the usual character orthogonality relations on the compact abelian group \(A\) give \[P_\chi P_{\chi'} = \begin{cases} P_\chi, & \chi=\chi',\\ 0, & \chi\neq \chi'. \end{cases}\]
Now suppose that \(F\in L^2(\Omega_i\times M)\) is orthogonal to every \(\mathcal{H}_\chi\). Then \[P_\chi F=0 \qquad\text{for every }\chi\in\widehat A.\] Consider the continuous positive-definite function \[c:A\to\mathbb{C}, \qquad c(a):=\langle U_aF,F\rangle_{L^2(\Omega_i\times M)}.\] For every \(\chi\in\widehat A\), its \(\chi\)-Fourier coefficient is \[\int_A c(a)\,\overline{\chi(a)}\,dm_A(a) = \left\langle \int_A U_aF\,\overline{\chi(a)}\,dm_A(a), F \right\rangle = \langle P_\chi F,F\rangle = 0.\] Since the characters of the compact abelian group \(A\) span a dense subspace of \(C(A)\), all Fourier coefficients of \(c\) vanish, and hence \(c=0\). In particular, \(0=c(e)=\|F\|_2^2\), so \(F=0\). Therefore the closed linear span of the spaces \(\mathcal{H}_\chi\), \(\chi\in\widehat A\), is all of \(L^2(\Omega_i\times M)\).
Applying this argument on each component \(\Omega_i\times M_i\) and summing over \(i\geq 0\), we obtain that \(L^2(Y)\) is the closed linear span of the \(k\)-vertical nilcharacters on \(\mathrm Y\). Pulling these functions back through the factor maps \(\mathrm X\to \mathrm Y_n\) proves the theorem.
Remark 8 (Relative orthonormality). The proof gives a slightly more structured statement on each bundle factor. Let \[\mathrm{Y}=\bigsqcup_{i\geq 0}\Omega_i\rtimes_{\alpha_i} M_i, \qquad M_i=N_i/\Gamma_i,\] be a bundle of nilsystems of step at most \(k\), and put \[A_i:=A_k(M_i)=(N_i)_k/((N_i)_k\cap\Gamma_i).\] For \(\chi\in\widehat{A_i}\), let \[\mathcal{K}_{i,\chi} := \Bigl\{ h\in L^2(M_i): h(a\cdot m)=\chi(a)h(m) \text{ for }m_{A_i}\otimes m_{M_i}\text{-a.e. }(a,m) \Bigr\}.\] The argument with the projections \(P_\chi\) shows that \[L^2(M_i)=\bigoplus_{\chi\in\widehat{A_i}}\mathcal{K}_{i,\chi}.\] Choose, for each \(i\) and each \(\chi\in\widehat{A_i}\), an orthonormal basis \((e_{i,\chi,\ell})_{\ell\geq 1}\) of \(\mathcal{K}_{i,\chi}\). Then the functions \[F_{i,\chi,\ell}(\xi,m) := 1_{\Omega_i}(\xi)e_{i,\chi,\ell}(m), \qquad \xi\in\Omega_i,\;m\in M_i,\] extended by zero outside \(\Omega_i\times M_i\), are \(k\)-vertical nilcharacters on \(\mathrm{Y}\).
Moreover, the family \((F_{i,\chi,\ell})_{i,\chi,\ell}\) is relatively orthonormal over the invariant factor \(\Omega\) in the following sense: \[\mathbb{E}\bigl(F_{i,\chi,\ell}\overline{F_{i',\chi',\ell'}}\mid\Omega\bigr)=0\] unless \((i,\chi,\ell)=(i',\chi',\ell')\), while \[\mathbb{E}\bigl(|F_{i,\chi,\ell}|^2\mid\Omega\bigr)=1_{\Omega_i}.\] Furthermore, the closed linear span of all products \[F_{i,\chi,\ell}\psi, \qquad \psi\in L^\infty(\Omega),\] is all of \(L^2(Y)\). Thus every bundle factor appearing in Theorem 2 admits a relative orthonormal basis of \(k\)-vertical nilcharacters, in direct analogy with the relative orthonormal basis of eigenfunctions of Frantzikinakis and Host [17].
Passing to the inverse limit in the proof of Theorem 3 therefore gives a countable family of non-ergodic \(k\)-vertical nilcharacters on \(\mathrm X\) whose \(L^\infty(\Omega)\)-linear span is dense in \(L^2(\mathrm Z_k(\mathrm X))\). Since the relative orthonormal bases obtained from different bundle factors in the inverse system need not be mutually relatively orthogonal, we only use this totality statement for the inverse limit.
Host and Kra [1] developed a theory of cubical structures and the corresponding seminorms. They used this framework to define systems of order \(k\) and proved a structure theorem showing that ergodic systems of order \(k\) are precisely \(k\)-step pro-nilsystems. This led to the identification of characteristic factors for multiple ergodic averages and to a proof of their convergence in \(L^2\). Independently, Ziegler [2] obtained the \(L^2\)-convergence of the same averages by proving that the universal characteristic factors she constructed are pro-nilsystems, via a different approach. Leibman [3] later clarified the connection between these two constructions, showing that the Host–Kra factors of order \(k\) coincide with Ziegler’s \(k\)th universal characteristic factors.↩︎