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On Weakly Almost Periodic Measures

We study the diffraction and dynamical properties of translation bounded weakly almost periodic measures. We prove that the dynamical hull of a weakly almost periodic measure is a weakly almost periodic dynamical system with unique minimal component given by the hull of the strongly almost periodic component of the measure. In particular the hull is minimal if and only if the measure is strongly almost periodic and the hull is always measurably conjugate to a torus and has pure point spectrum with continuous eigenfunctions. As an application we show the stability of the class of weighted Dirac combs with Meyer set or FLC support and deduce that such measures have either trivial or large pure point respectively continuous spectrum. We complement these results by investigating the Eberlein convolution of two weakly almost periodic measures. Here, we show that it is unique and a strongly almost periodic measure. We conclude by studying the Fourier-Bohr coefficients of weakly almost periodic measures.


Introduction
Over the past 100 and more years physical diffraction gave us valuable insight into the atomic structure of solids. Periodic crystals produce a clear diffraction pattern, consisting exclusively of bright spots (Bragg peaks) positioned on a lattice (the dual lattice of the periods lattice of the crystal).
While often the atomic structure of the crystal can be reconstructed from the diffraction pattern, this is not always the case: there exists different periodic crystals with the same diffraction pattern. The problem becomes even more complex in the case of quasicrystals, or more generally, systems with mixed spectrum (see e.g. [24] for a recent overview of the inverse problem).
Mathematically, physical diffraction is described as follows: starting from the underlying structure, which is modeled by a discrete set or more generally by a measure, we construct a measure γ, called the autocorrelation measure, which encodes the frequencies of vectors between atoms in the structure. This measure is positive definite, therefore Fourier transformable with a positive Fourier transform p γ [2,12,40]. The positive measure p γ is called the diffraction measure or diffraction pattern and models the physical diffraction of the original solid. The Lebesgue decomposition p γ " p γ d`p γ c gives us a decomposition of the diffraction into the discrete or pure point part p γ d and the continuous part p γ c . (The continuous part can then be further split into the absolutely continuous part and the singularly continuous part.) The pure point component p γ d can be written as where B is the set of Bragg peaks. As any positive definite translation bounded measure, the autocorrelation is a weakly almost periodic measure [28,40]. The Eberlein decomposition γ " γ s`γ0 , into the strong and null weakly almost periodic part is the Fourier dual of the above mentioned Lebesgue decomposition of p γ into the discrete and continuous part [40,28] or [45,Thm. 2.8]. This implies in particular that γ is strongly almost periodic if and only if p γ is a pure point measure. Recently there is also increased interest in diffraction of systems which are described by more general structures than measures [11,32,49,50]. In particular, [11,49,50] deal with tempered distributions. Also, in these cases we end up with weakly almost periodic distributions and there exists also an Eberlein decomposition of tempered distributions which is dual to the Lebesgue decomposition [48].
Almost periodicity has long been known to play a role in the study of pure point diffraction, see e.g. the paper [43] of Solomyak for an early explicit investigation. The possible connections between almost periodicity properties of the original point set or measure and pure point diffraction were then the topic of a survey article of Lagarias [27], which contains many questions and conjectures on this topic. The actual statement on the equivalence of pure point diffraction and strong almost periodicity of the autocorrelation measure discussed above was first given explicitly in the work of Baake-Moody [10]. The main thrust of their work concerns the case of measures with Meyer set support. In that case, strong almost periodicity of the autocorrelation can even be replaced by the stronger notion of norm almost periodicity [10]. Connections of strong almost periodicity of measures and pure point diffraction were then also studied by Moody-Strungaru [36] and Gouéré [23] (see also Favarov [20] and Kellendonk-Lenz [25] for related material discussing strongly almost periodic point sets). Also, the connection between almost periodicity and the cut and project formalism thoroughly investigated by Baake-Moody [10], Richard [38], Lenz-Richard [33] and Strungaru [47]. As shown recently in Lenz [31] it is even possible to extend parts of the above theory from dynamical systems based on measures to general dynamical systems allowing for a metric.
All these results emphasize that the class of (weakly) almost periodic measures is important for diffraction theory. It is our goal in this paper to systematically study the long range order properties of this class via their dynamical system and the diffraction measure.
As far as methods and tools go the paper builds up on two related theories. On the one hand it can be seen as a continuation of the work of Eberlein [16,17,18] on weakly almost periodic functions and measures in the context of pure point diffraction. On the other hand it can be seen as an application of the theory of weakly almost periodic systems to a specific class of measure dynamical systems. This theory was started by Ellis and Nerurkar [19] and has later been extended in various works, see e.g. the recent work of Akin-Glasner [1], where various further references can be found.
The paper is organized as follows: we start by reviewing basic concepts in harmonic analysis on locally compact abelian groups in Section 1. There, we also introduce one of the main players of our investigation viz the autocorrelation of a measure. We will be interested in the autocorrelation of weakly almost periodic measures and much of our study is built on dynamical systems. The corresponding background dynamical systems, measure dynamical systems and on almost periodic measures and corresponding topologies on the set of all translation bounded measures is discussed in Section 2 This material is essentially all known. We discuss it at some length in order to keep the present paper accessible to a wide audience.
Our actual investigation starts in the subsequent Section 3. There, we present a thorough analysis of the hull of a weakly almost periodic measure. In particular, we show in Section 3.1 that the hull is a weakly almost periodic dynamical system. In the subsequent Section 3.2 we discuss how the classes of weakly, null-weakly and strongly almost periodic measures are stable under taking the hull. We show that the hull of a weakly almost periodic measure has a unique minimal component in Section 3.3. In this context, a main result of the paper, Theorem 3.15, is proven. This result shows that for every weakly almost periodic measure µ, its strongly almost periodic component µ s belongs to the hull Xpµq. Most of the subsequent considerations rely on this result. One implication is that the hull generated by µ s is the unique minimal component. In fact, based on this result we can discuss the structure of the hull of a weakly almost periodic system in quite some detail in the subsequent Section 3.4. There, we show that the projections P s , P 0 onto the strongly and the null-weakly almost periodic part are continuous G-mappings with range contained inside Xpµ s q respectively Xpµ 0 q.
In Section 4 we show that the dynamical system of a weakly almost periodic measure µ has pure point dynamical and diffraction spectra. Here, we also provide a closed formula for the eigenfunctions of the system, which are shown to be continuous, and for the diffraction measure. When combining this with Theorem 3.12 from Section 3.2 we infer as a remarkable feature of our setting that the eigenfunctions relevant for diffraction are just given by the Fourier coefficients of the measure in question.
The preceding results on weakly almost periodic measures have an application to arbitrary measure dynamical systems. The reason is that each measure dynamical system gives rise to an autocorrelation which is weakly almost periodic. It turns out that the diffraction of the dynamical system arising as the hull of the autocorrelation is just the pure point part of the diffraction of the original system. This is discussed in Section 5.
Our results have various application to Delone sets. This is discussed in Section 6. There, we first relate in Section 6.1, for weakly almost periodic weighted Dirac combs µ with FLC support, the support of µ s and of µ 0 to the support of µ. We then use our results to give rather direct alternative proofs to recent results on the support of the Eberlein decomposition for weighted Dirac combs. This is discussed in Section 6.2.
We complete the paper by looking at the Eberlein convolution of weakly almost periodic measures in Section 7. We show that given any two weakly almost periodic measures they have an unique Eberlein convolution, which exists with respect to any van Hove sequence and is strongly almost periodic. As a consequence we get (an alternative proof) that any weakly almost periodic measure has pure point diffraction and hence pure point dynamical spectrum.
By combining weakly almost periodic dynamical systems and aperiodic order in our paper we emphasize that there is a deep connection between these concepts, which could be of interest to mathematicians from both areas. In order to keep the paper accessible for both communities we have taken special care to discuss basic concepts at sufficient length.

General background and notation
For this entire paper G denotes a locally compact Abelian group (LCAG). We will denote by C u pGq the space of uniformly continuous and bounded functions on G, and by C c pGq the subspace of C u pGq consisting of functions with compact support.
For a compact set K Ă G we define CpG : Kq :" tf P C c pGq| supppf q Ă Ku .
The space pCpG : Kq, } } 8 q is a Banach space. On C c pGq we define a locally convex topology as the inductive limit topology defined by the embeddings CpG : Kq ãÑ C c pGq, for K Ă G compact.
For a function f on G we define the functions f : and r f on G by Similarly, for a measure µ we define a new measure r µ by r µpf q :" µp r f q .
By Riesz-Markov theorem, a measure µ on G can be viewed as a linear functional on C c pGq, which is continuous with respect to the inductive topology on C c pGq, see [41,37] or [39,Appendinx] for details. We will refer to the weak-* topology of this duality as the vague topology for measures.
Any measure µ gives rise to an unique positive measure |µ|, called the variation measure of µ, satisfying |µ| pf q " supt|µpgq| : g P C c pG : Rq, with |g| ď f u , for any non-negative f P C c pGq. We can define the convolution between c P C c pGq and g P C u pGq as c˚gpxq " ż G cpx´tqgptqdt .
Then c˚g P C u pGq. Similarly, the convolution between a function c P C c pGq and a measure µ is the function c˚µpxq " ż G cpx´tqdµptq .
We will restrict attention to a class of measures which are "equi-bounded" on G: where |µ| denotes the variation of µ. The space of all translation bounded measures on G is denoted by M 8 pGq.
To check that a measure is translation bounded, it is sufficient to check that (1) holds for one fixed compact set K with non-empty interior [10]. Also, the translation boundedness property of measures can be understood via convolutions with compactly supported continuous functions. Indeed, by [2, Thm.1.1] a measure µ is translation bounded if and only if for all c P C c pGq we have c˚µ P C u pGq. The convolution of function can be extended to measures the following way: We say that the measures µ and ν are convolvable, if for all f P C c pGq the function px, yq Ñ |f px`yq| is in L 1 p|µ|ˆ|ν|q. In this case, we define the convolution µ˚ν by In general for translation bounded measures we can try to also define an averaged convolution (Eberlein convolution) of the measures. This point will be taken up and discussed extensively later on in the last section of the article. Here, we introduce next what makes a good averaging set. Definition 1.2. A sequence pA n q of compact subsets of G called a van Hove sequence if for all compact sets K Ă G we have where the K-boundary is defined by: A sequence pA n q of compact subsets of G is called a Fölner sequence if for all It is easy to see that any van Hove sequence is a Fölner sequence.
We are now ready to introduce the notion of autocorrelation measure. Definition 1.3. Let µ P M 8 pGq be a measure and let pA n q be a van Hove sequence. We say that µ has the autocorrelation γ with respect to pA n q if the following vague limit exists: γ :" lim where µ| An denotes the restriction of the measure µ to the set A n . (Here, the convolution makes sense as the measures µ| An and Ć µ| An are finite due to the compactness of the A n .) Note that existence of the limit in the definition is rather a matter of convention. Indeed, the limit will always exist for a suitable subsequence of pA n q. If the limit exists for all van Hove sequences pA n q, it must be the same for all. In this case we will say that µ has an unique autocorrelation.
The autocorrelation has an extra positivity property which we introduce next: It is well known (see e.g. [12]) that a measure µ is positive definite if and only if for all f P C c pGq the function µ˚pf˚r f q is positive definite.
It is also easy to see that for any finite measure ν, the measure ν˚r ν is positive definite, and that vague limits of positive definite measures are positive definite. Thus, we immediately obtain the following corollary.
Corollary 1.5. Let γ be the autocorrelation of µ with respect to the van Hove sequence pA n q. Then, γ is positive definite.

Measure dynamical systems and weakly almost periodic measures
The use of dynamical systems in the study of long range order has a long history, see e.g. the survey [42] for an early account. As in statistical mechanics, the basic idea is to look at ensembles rather than at individual manifestations. More specifically, the idea is to construct out of a given point set Λ Ă G, the collection XpΛq of all point sets which locally look like Λ. This collection becomes compact with respect to the local rubber topology, and comes with a natural G-action, therefore it becomes a topological dynamical system. The diffraction of Λ can then be connected to the dynamical spectrum of XpΛq via the Dworkin argument [7,23,30,15], and all these ideas can be generalized to measures [7,34]. We recommend [8] for a recent review of this.
2.1. Measure dynamical systems. We will be interested in dynamical systems coming from translation bounded measures.
Whenever X is a compact space equipped with a continuous action T : GˆX ÝÑ X of the group G, we call pX, T q a dynamical system. Such a system is called minimal if the orbit tT t x : t P Gu of x P X is dense in X for every x P X. If there exists one dense orbit tT t x : t P Gu then the system is called transitive. A system is called uniquely ergodic if there exists only one G-invariant probability measure on X. A special focus of ours will be on continuous eigenfunctions. Here, an f P CpXq with f ‰ 0 is called an eigenfunction to χ P p G if f pT t xq " χptqf pxq holds for all t P G and x P X. Such a χ is called a continuous eigenvalue. We will be interested in special transitive dynamical systems coming from measures and more specifically coming from weakly almost periodic measures. In this context we define for a translation bounded measure µ on G the measure T t µ by where δ t denotes the unique point mass at t P G.
It is not hard to see that the translation action is continuous when restricted to compact G-invariant subsets of M 8 pGq. So any such compact G-invariant X is a dynamical system pX, T q. We refer to such a dynamical system as measure dynamical system. In particular, any µ P M 8 pGq gives rise to a dynamical system Xpµq defined as where the closure is taken in the vague topology. This dynamical system is called the hull of µ. We recommend [7] for more details.
A crucial feature of any measure dynamical system pX, T q is that there is a linear map φ from C c pGq to the set C c pXq of continuous functions on X. More specifically, each c P C c pGq defines a function φ c : These functions are continuous [7], and in fact the vague topology on M 8 pGq is the weakest topology which makes these functions continuous [7]. When dealing with a given measure dynamical system we can (and will tacitly) restrict these functions to the dynamical system. Accordingly, these functions will appear often in our computations. We will also make often use of φ c pT t νq " c˚νptq.

2.2.
Weakly almost periodic functions and measures. In this section we review the basic definitions and properties of weakly almost periodic functions and measures.
We start by recalling the notions of almost periodicity we will use. Here, for a function f on G and t P G we define the translate T t f of f by t to be the function with Definition 2.1. A function f P C u pGq is called Bohr-almost periodic if the set tT t f |t P Gu is precompact in pC u pGq, } } 8 q. The set of all Bohr-almost periodic function on G is denoted by SAP pGq.
A function f P C u pGq is called weakly almost periodic if the set tT t f |x P Gu is precompact in the weak topology of the Banach space pC u pGq, } } 8 q. The space of all weakly almost periodic functions is denoted by W AP pGq.
We summarize now the basic structural stability properties of the spaces W AP pGq as proven in [16] and of SAP pGq as discussed recently in [40]. (e) If f P SAP pGq and p ě 0 then |f | p P SAP pGq.
(f) If f P W AP pGq and p ě 0 then |f | p P W AP pGq.
For our further investigation we will also need the following fundamental fact.

Theorem 2.3. [16]
If f P C u pGq is a positive definite function on G then f belongs to W AP pGq.
Next we recall the notion of amenability for continuous functions. Definition 2.4. Let pA n q be a Fölner sequence. A function f P C u pGq is called amenable if there exists a number M pf q, called the mean of f such that Remark. [16] The definition of amenability and the mean are independent of the choice of the Fölner sequence.
The basic properties of the mean M : W AP pGq Ñ C are summarised below.
Theorem 2.5. [16, Thm. 14.1] Any weakly almost periodic function f is amenable. If f, g P W AP pGq, C 1 , C 2 P C and t P G then we have: Moreover, if L : W AP pGq Ñ C is a function which satisfies (a), (b), (c), (d) and (f ) then Lpf q " M pf q for all f P WAPpGq.
As a consequence of Theorem 2.2 and Theorem 2.5 we note the following.
We are now ready to introduce the notion of null-weakly almost periodicity.
Definition 2.7. A function f is called null weakly almost periodic if f P W AP pGq and M p|f |q " 0. We will denote the space of null weakly almost periodic functions by W AP 0 pGq.
The relevance of the space of null-weakly almost periodic functions comes from the following result.
The theorem says that any weakly almost periodic function f can uniquely be written as the sum f " f s`f0 , of a strongly almost periodic function f s and a weakly almost periodic function f 0 .
We will refer to this decomposition as the Eberlein decomposition of weakly almost periodic functions.
Next we see the Eberlein convolution of functions: where M t denotes the mean with respect to the variable t. The function f f g is called the Eberlein convolution of f and g. (For a fixed x P G we have f px´¨qgp¨q P W AP pGq and therefore the mean exists.) The above concepts can be extended to measures via convolutions. This has been carried out by deLamadrid and Argabright [28] (see also [40]): Definition 2.11. A translation bounded measure µ is called strongly, weakly respectively null weakly almost periodic if for all c P C c pGq the function c˚µ is Bohr, weakly respectively null weakly almost periodic. We denote the spaces of weakly, strongly and null weakly almost periodic measures by WAPpGq, SAP pGq respectively WAP 0 pGq.
Exactly as for functions, we can define in a simple way the mean of a weakly almost periodic measure.
Proposition 2.12. [28,40] Let µ be a weakly almost periodic measure. Then, there exists a constant M pµq such that, for all c P C c pGq we have Definition 2.13. Given a measure µ P WAPpGq, we call the number M pµq from Proposition 2.12, the mean of µ.
The mean of a measure can be calculated by a formula which is similar to the formula for functions.
Lemma 2.14. [40] Let µ P WAPpGq and pA n q a van-Hove sequence on G. Then The Eberlein decomposition can then be extended to measures: As for functions, we will refer to this decomposition as the Eberlein decomposition.
Let us emphasize here that the space of null weakly almost periodic measures is more enigmatic than the space of null weakly almost periodic functions: Consider a weakly almost periodic measure µ. Then, by definition, it is null weakly almost periodic if and only if for all c P C c pGq we have M p|c˚µ|q " 0. However, this is not equivalent to M p|µ|q " 0 (even if the variation measure |µ| is weakly almost periodic). Indeed, as proven in [44,47], for weakly almost periodic measures with uniformly discrete support we have the equivalence µ P WAP 0 pGq if and only if M p|µ|q " 0, but in general we only have the implication M p|µ|q " 0 ñ µ P WAP 0 pGq. The measure is a null weakly almost periodic measure, but M p|µ|q " 2, see [44].
Next we introduce the Fourier Bohr coefficients of a weakly almost periodic measure. Whenever f is strongly almost periodic function and µ is a weakly almost periodic measure we have f µ P WAPpGq[28, Thm. 6.2]. We can thus define.
Definition 2.16. The Fourier Bohr coefficients of a weakly almost periodic measure are defined for each character χ P p G as Similarly, if f P W AP pGq we define the Fourier Bohr coefficients of f as It is easy to see that c χ pf q " c χ pf θ G q.
As proven in [28, Thm. 8.1], the null weakly almost periodic measures/functions are exactly the weakly almost periodic measures/functions with vanishing Fourier -Bohr coefficients. It follows from the uniqueness of the Eberlein decomposition that a strongly almost periodic measure/function is uniquely determined by its Fourier Bohr coefficients.
The Fourier Bohr coefficients of the Eberlein convolution are exactly the products of the Fourier Bohr coefficients of the two functions. This result is proven in [16,Lemma 1.15] in the particular case g " r f and in [28] in the case of a convolution between a strong almost periodic function and a weakly almost periodic measure. For completeness reasons we include a proof (which is the standard one) for the general case. Then

Proof. A direct computation gives
Now, by the Fubini Theorem for the mean [16, Thm.14.2], we have As a χ pf q is a constant, the claim follows.

2.3.
Product and weak topology on M 8 pGq and WAPpGq. In this section we provide another perspective on the weak and strong almost periodic measures by considering two further topologies on M 8 pGq.
Consider the natural embedding Via this embedding the product topology on rC u pGqs CcpGq induces a topology on M 8 pGq. This topology is called the product topology for measures. Let us note that the product topology is a locally convex topology defined by the family of seminorms t} } c u cPCcpGq given by Thus, it also allows for a corresponding weak topology. We will refer to this weak topology as the weak topology for measures. Here comes a characterization of (weak) almost periodicity of measures via the previously introduced topologies.  We complete the section by showing the continuity of the functions c χ on WAPpGq. We start with a standard lemma from topology. Lemma 2.20. [29] Let τ 1 and τ 2 be topologies on X such that τ 1 is stronger than τ 2 and τ 2 is a Hausdorff topology. If A Ă X is a compact set with respect to τ 1 then A is compact with respect to τ 2 and the two topologies coincide on A.
Theorem 2.21. Let χ P p G. Then, the Fourier coefficient c χ : WAPpGq ÝÑ C is weakly continuous on WAPpGq. In particular, if K is any weakly compact subset of WAPpGq, then c χ is vaguely continuous on K.
Proof. First, let us observe that we have This shows that c χ is continuous on pW AP pGq, } } 8 q. Then, by the Hahn Banach Theorem, we can extend this to a continuous functional f χ on pC u pGq, } } 8 q. Furthermore, by the definition of the product topology, for each c P C c pGq the mapping pM 8 pGq, product topologyq Ñ pC u pGq, } } 8 q, µ Ñ c˚µ is continuous from M 8 . It follows that this mapping is weakly continuous, see e.g. [40]. Therefore, the composition µ Ñ f χ pc˚µq is a weakly continuous mapping on M 8 , and hence its restriction µ Ñ c χ pc˚µq is weakly continuous on WAPpGq.
Finally, let us pick some c P C c pGq with p cpχq ‰ 0. Then, as we get the weak continuity of c χ on WAPpGq.
The last statement follows as by Lemma 2.20 and Proposition 2.18, the vague and weak topologies coincide on K.
Remark. When we extended c χ to a continuous linear functional on C u pGq in order to get the weak continuity of c χ we have actually seen a very particular case of the following general phenomena, which is an immediate consequence of the Hahn-Banach Theorem: If pE, } }q is a Banach space, and F is a closed subspace of E, then the weak topology of pF, } }q is the same as the inherited weak topology on F as a subspace of E.

The hull of a weakly almost periodic measure
In this section we present a rather thorough study of the hull of a weakly almost periodic measure.
3.1. The hull is a weakly almost periodic system. In this section we provide an important structural insight concerning the hull of a weakly almost periodic measure: This hull is a weakly almost periodic dynamical system. This makes a wealth of results available to the study of weakly almost periodic measures.
Whenever pX, T q is a dynamical system a function f P CpXq is called weakly almost periodic if the set tf˝T t : t P Gu is relatively compact in the weak topology of pCpXq, }¨} 8 q. We denote by W AP pXq the subset of CpXq consisting of weakly almost periodic functions. A dynamical system pX, T q is called weakly almost periodic if every f P CpXq is an weakly almost periodic function, or equivalently if CpXq " W AP pXq.
In order to prove the main result of this section we next carry out a study of W AP pXq for general dynamical systems. As is will appear in many proofs, we define weak the hull of f by

.8])
If Ω is a compact space, and f n , f P CpΩq, then f n Ñ f weakly if and only if }f n } 8 is bounded and f n Ñ f pointwise.
Next, we will often make use of the following classical result, which allows us to use sequences instead of nets to prove weakly almost periodicity.  (1)] Let f P CpXq. Then f P W AP pXq if and only if for each sequence pt n q in G we can find a subsequence pt kn q such that f˝T t kn is weakly convergent to some g P CpXq.
We are now ready to prove a few useful results about the space W AP pXq. Note that the proofs are almost identical to the ones from [16,40], but since those in the cited papers dealt only with W AP pGq, we are in a completely different situation. Nevertheless the basic ideas and tools remain the same. Proof. Let f, g P W AP pXq, and let pt n q be any sequence in G. By f P W AP pXq there exists by Corollary 3.2 a subsequence pt kn q such that f˝T t kn is weakly convergent to some f 1 P CpXq. Similarly, by g P W AP pXq, there exists again by Corollary 3.2 a subsequence pt ln q of pt kn q such that g˝T t ln is weakly convergent to some g 1 P CpXq.
As f˝T t ln respectively g˝T t ln converge weakly to f 1 and g 1 , respectively, by Lemma 3.1, f˝T t ln respectively g˝T t ln are equi-bounded and converge pointwise to f 1 and g 1 , respectively. Then, it is immediate to see that pf`gq˝T t ln respectively pf¨gq˝T t ln equi-bounded and converge pointwise to f 1`g1 and f 1¨g1 , respectively. It now follows from Corollary 3.2 that f`g, f¨g P W AP pXq.
It is trivial to check that W AP pXq is closed under scalar multiplication, and it contains the constant function 1. Proof. Since pCpXq, } } 8 q is a Banach space, it suffices to work with sequences.
Let pf n q be a sequence in W AP pXq such that, in the sup norm, f n Ñ f P CpXq. We need to show that f P WAPpXq. We prove this by using Corollary 3.2: Let pt m q in G be arbitrary. Then, we can pick some subsequence pt 1 m q of pt m q such that f 1˝T t 1 m is weakly convergent to some g 1 P CpXq. By induction, for each n ě 2 we can pick some subsequence pt n m q of pt n´1 m q such that f n˝T t n m is weakly convergent to some g n P CpXq. Consider now the diagonal s m :" t m m , m P N. Then, for all n we have w´lim m f n˝T sm " g n .
We show next that g n is Cauchy in pCpXq, } } 8 q. Indeed Therefore, as pCpXq, } } 8 q is complete, the sequence pg n q converges to some g P CpXq.
We claim now that f˝T sm converges weakly to g. This claim, once proven, completes the proof by Corollary 3.2.
Let ǫ ą 0 and ψ P CpXq 1 . Since g n Ñ g and f n Ñ f , there exists some N such that for all n ą N we have Pick some n 0 ą N . As w´lim m f n0˝T sm " g n0 there exists some M such that for all m ą M we have |ψpf n0˝T sm´g n0 q| ă ǫ . Then, for all m ą M we have |ψpg´f˝T sm q| ď |ψpg´g n0 q|`|ψpg n0´fn0˝T sm q|`|ψpf n0˝T sm´f˝T sm q| This proves the claim.
We complete our considerations on general dynamical systems by proving the following result. Part (a) can also be found in [22, Thm. 1.43 (4)] Theorem 3.5. Let pX, Gq be a dynamical system. Let f P CpXq and ω P X be given and define g : G Ñ C via gptq " f pT t ωq. Then, the following holds: (a) If f P W AP pXq then g P W AP pGq. (b) If g P W AP pGq and tT t ω | t P Gu is dense in X then f P W AP pXq.
Proof. Before proving (a) and (b) we need a little preparation. Define It is easy to see that F is well defined. Indeed, as X is compact, h is bounded and hence so is F phq. Moreover, h is uniformly continuous, and by the continuity of the group action on X so is F phq. Moreover, (2) }F phq} 8 " sup sPG |hpT s pωqq| ď }h} 8 .
Let us also observe that if tT t ωt P Gu is dense in X then we have equality in (2). Now we proceed to prove (a) and (b).
(a) F is a continuous operator between the Banach spaces pCpXq, } } 8 q and pC u pGq, } } 8 q and hence it is clearly weakly continuous (see also [40] for a recent discussion of this type of reasoning). Since Wpf q is weakly compact, and F is weakly continuous, it follows that F pWpf qq is weakly compact in C u pGq. Now, by construction we have T t F pf q " F pT´tf q. Therefore tT t F pf q|t P Gu Ă F pWpf qq , and hence g " F pf q P W AP pGq.
(b) Since tT t ωt P Gu is dense in X, we have equality in (2). Therefore, F gives an isometric embedding of CpXq ãÑ C u pGq. Denote by Y :" tF phq|h P CpXqu Ă C u pGq .
Then, F is an isometry between CpXq and Y. Since CpXq is a Banach space, so is Y and therefore Y is closed in C u pGq. As F is linear, Y is a subspace of C u pGq. As a closed subspace Y is also weakly closed in C u pGq.
Consider now the isometry F´1 : Y Ñ CpXq. Since As Y is weakly closed, and g P W AP pGq it follows that weak closure tT t g|t P Gu w of the orbit of g is compact and contained Y. Then, as F´1 is continuous, it is weakly continuous, and therefore F´1pJq is weakly compact. As this set contains the orbit of f , we get that f P W AP pXq.
We can now come to the main result of this section.
Theorem 3.6. Let µ P M 8 pGq be given. Then, µ is weakly almost periodic if and only if pXpµq, T q is a weakly almost periodic dynamical system.
Proof. This follows from Theorem 3.5 applied with the functions φ c : M 8 pGq ÝÑ C, φ c pµq " c˚µp0q for c P C c pGq. Note that φ c pT t µq " c˚µptq. Here are the details: Assume that pXpµq, Gq is weakly almost periodic. Let c P C c pGq. Since φ c P W AP pXpµqq it follows from Theorem 3.5 (a) that c˚µ P W AP pGq. As this is true for all c P C c pGq we get µ P WAPpGq.
Assume conversely that µ is a weakly almost periodic function. If c P C c pGq then it follows from Theorem 3.5 (b) that φ c P W AP pXpµqq. This shows that φ c P W AP pXpµqq for all c P C c pGq. Then, by Proposition 3.3, W AP pXpµqq contains the algebra generated by tφ c |c P C c pGqu. Since this algebra is dense in CpXpµqq [7] and since W AP pXpµqq is a closed subspace of pCpXpµqq, } } 8 q, it follows that This finishes the proof.
We now derive a few corollaries from this concerning strongly almost periodic systems and measures. Here, a dynamical system pX, T q is called strongly almost periodic if for any f P CpXq the set tf˝T t : t P Gu is relatively compact in pCpXq, }¨} 8 q. We will need the following well-known statements on strongly almost periodic systems. (b) For a transitive dynamical system pX, Gq with dense orbit of ω P X the following statements are equivalent: (i) pX, Gq is strongly almost periodic.
(ii) pX, Gq admits a structure of a locally compact group such that G ÝÑ X, t Þ Ñ T t ω becomes a continuous group homomorphism. (iii) pX, Gq is weakly almost periodic and minimal.
Proof. (a) By [19,Prop. 2.8] any minimal weakly almost periodic dynamical system is strongly almost periodic. Now, (a) follows as any minimal component of a weaklyalmost periodic system is also weakly almost periodic. (Indeed, if Y is an invariant compact subset of X then any f P CpYq can be extended by compactness to some g P CpXq. Define now R : CpXq Ñ CpYq to be the restriction of the domain to Y. Then we have }R} ď 1, which shows that R is continuous, hence weakly continuous. Therefore, as g P CpXq " W AP pXq we have f " Rpgq P W AP pYq.) (b) The equivalence of (i) and (ii) can be found e.g. in the book of Auslander [3]. The implication (iii)ùñ (i) is immediate from (a). On the other hand (i) clearly implies weak almost periodicity and (ii) clearly implies minimality of pX, Gq and so (iii) follows from (i) and (ii).
Here are the corollaries. Proof. As is well-known, see e.g. [33] for a recent discussion, pXpµq, Gq admits a group structure such that t Þ Ñ T t µ is a continuous group homomorphism if and only if µ is strongly almost periodic. Now, the statement follows from the equivalence of (i) and (ii) in (b) of Proposition 3.7. Corollary 3.9. Let µ P WAPpGq. Then, Xpµq is minimal if and only if µ P SAPpGq.
Proof. By Theorem 3.6 pXpµq, Gq is weakly almost periodic. Hence, by the equivalence between (i) and (iii) in (b) of Proposition 3.7, it is strongly almost periodic if and only if it is minimal. Now, the desired statement follows from the preceding corollary.

3.2.
Stability of almost periodicity along the hull. In this section we show that all elements in the hull of a strongly, weakly and null-weakly almost periodic measure are strongly, weakly and null-weakly almost periodic respectively.
We begin with the following well-known consequence of Lemma 2.20. Lemma 3.10. Let τ 1 and τ 2 be topologies on X such that τ 1 is stronger than τ 2 and τ 2 is Hausdorff. If A Ă X is precompact with respect to τ 1 , then the closures of A with respect to τ 1 and τ 2 coincide, and the two topologies coincide on this closure.
Here is the first stability result of this section.
(a) The following assertions are equivalent: The hull Xpµq agrees with the closure of tT t µ | t P Gu in the weak topology and is, in particular, compact in the weak topology. (iii) The vague and the weak topologies agree on Xpµq.
(b) The following assertions are equivalent: The hull Xpµq agrees with the closure of tT t µ | t P Gu in the product topology and is, in particular, compact in the product topology. (iii) The vague and the product topology agree on Xpµq.
Proof. (a) (i)ùñ (ii): As µ is weakly almost periodic, the closure of its orbit in the weak topology is compact. Now, (ii) follows from the previous Lemma as the weak topology is stronger than the vague topology.
(iii)ùñ (iv): Let ν P Xpµq be arbitrary. We then have Xpνq Ă Xpµq. As the vague and the weak topology agree on Xpµq we infer that Xpνq is compact in the weak topology. Hence, ν belongs to WAPpGq. This gives (iv).

(b) This can be shown by analogous arguments.
We next show that for weakly almost periodic measures, the Fourier Bohr coefficients depend continuously on the measure. Theorem 3.12. Let µ P WAPpGq and χ P p G. Then c χ : Xpµq Ñ C is continuous and satisfies c χ pT t νq " χptqc χ pνq .
In particular, if c χ pµq ‰ 0, then c χ is a continuous eigenfunction of Xpµq.
Proof. Since µ P WAPpGq the hull Xpµq is weak compact. The continuity of c χ follows now from Theorem 2.21. Moreover, for all t P G we have c χ pT t νq " M pχT t νq " M pT t`p T´tχqν˘q " M pχptqpχqνq " χptqc χ pνq and the proof is finished.
Remark. As we will see later, the intensity of the Bragg peak at χ P p G in the diffraction of µ is given by |c χ pµq| 2 . So Theorem 3.12 gives that each Bragg peak χ, comes from a continuous eigenfunction on Xpµq. Proof. This follows easily from the previous theorem and the fact that a weakly almost periodic measure is null weakly almost periodic if and only if all of its Fourier coefficients are zero [28, Thm. 8.1]. Specifically, as µ is null weakly almost periodic all of its Fourier coefficients are zero. By the formula given in the previous theorem this then holds as well for all elements in the orbit of µ and by the continuity statement of the previous theorem, this then holds for the whole hull.
3.3. The unique minimal component and unique ergodicity. In this section we show that the hull of a weakly almost periodic measure has a unique minimal component and that this component is just the hull of the strongly almost periodic part of the measure. This has strong consequences. For example a weakly almost periodic dynamical system is uniquely ergodic if and only if it has a unique minimal component.
A first step (and main result in this section) is that the strongly almost periodic part µ s of µ belongs to the hull. As a consequence we can then derive a wealth of results on weakly almost periodic measures in this section and the next one.
As a preparation of the proof of the main result of this section we show that for a uniformly continuous function f , whose average absolute value integral is 0, we can find arbitrarily large regions in G where |f | is arbitrarily small. Lemma 3.14. Let pA n q be a van Hove sequence. Let f be an uniformly continuous bounded function on G such that lim n ş An |f px`tq| dx θ G pA n q " 0 .
uniformly in t P G. Then for all compact sets K Ă G and all ǫ ą 0, there exists some t P G so that |f pxq| ă ǫ for all x P t`K.
Remark. Note that the lemma applies to any null weakly almost periodic function.
Proof. Assume the contrary. Then, there exists a compact K and an ε ą 0 such that for any t P G there exists an x t P K with |f pt`x t q| ě ε. As f is uniformly continuous, there exists then an open relatively compact set U with |f pt`yq| ě ε 2 for all t P G and all y P x t`U . This implies, in particular, ż K`U |f pt`xq|dx ě ε 2 θ G pU q ": c ą 0 for all t P G and, hence, On the other hand, from the assumption on f and as pA n q is van Hove we have ş An |f pt`xq|dx θ G pA n q Ñ 0, n Ñ 8 uniformly in t P G.
From Fubini-theorem, we then obtain ş An ş K`U |f pt`xq|dxdt θ G pA n q " ż K`U˜ş An |f pt`xq|dx θ G pA n q¸d t Now, by the uniform convergence of ş An |f pt`xq|dx θGpAnq , there exists some N such that, for all n ą N we have ş An |f pt`xq|dx θ G pA n q ă c 2θ G pK`U q .
Then, for all n ą N we have ş which contradicts (3). This completes the proof.
Remark. Let us shortly discuss the assumptions in the lemma: The uniform continuity of f is essential in Lemma 3.14, as can be seen by considering the following example: For each n P Z pick some f n P C c pRq such that f n ě 0, f n pnq " 1, supppf n q Ă pn´1 2 , n`1 2 q and Then f is non-negative and has finite integral. Hence, it has zero average integral. However, clearly any translate of r´1, 1s will meet an element where f takes the value 1.
The assumptions of boundedness of f and of uniform existence of the limit can be removed if vanishing of the limit is known for any van Hove sequence.
We are now ready to prove the main result of this section viz that the hull of a weakly almost periodic function contains its strong almost periodic part. In particular, Xpµ s q Ă Xpµq.
Proof. We need to show that for every natural number n and all c 1 , . . . , c n P C c pGq and ǫ ą 0 there exists some r P G so that for all 1 ď i ď n we have |pT r µqpc i q´µ s pc i q| ă ǫ .
The basic idea of the proof is now to chose a compact K so that any translate of K contains a vector r such that T r µ s is very close to µ s (which is possible by almost periodicity of µ s ) and then to shift this K so that T r µ 0 pc i q is very small for all r in the shifted K (which is possible by the previous lemma).
Here, are the details: Let c 1 , ..., c n P C c pGq and ǫ ą 0 be given. Since the functions µ s˚c : i are almost periodic, the set P :" tt P G| max 1ďiďn }µ s˚c is relatively dense. Fix a compact K such that 0 P K and P`K " G. Define Then g 0 is a null weakly almost periodic function. By Lemma 3.14, there exists then some t P G so that |g 0 pxq| ă ǫ 2 for all x P t´K. Since t P G " P`K, there exists some r P P and k P K such that t " r`k. Therefore and hence, by the definition of g 0 , for all 1 ď i ď n we havěˇˇµ 0˚c : Moreover, as r P P we also have Combining these two relations we get:ˇˇµ s˚c : i p0q´rT r µ s`T r µ 0 s˚c : i p0qˇˇă ǫ . Therefore |µ s pc i q´rT r µspc i q| ă ǫ .
which completes the proof. Proof. By definition µ s is strongly almost periodic. Hence, its hull Xpµ s q is minimal. Moreover, by the previous theorem, this hull is part of Xpµq. It remains to show the uniqueness part of the statement: Let ν be any element of Xpµq belonging to a minimal component. Then, it is strongly almost periodic by abstract principles (see above). Hence, ν " ν s . Now, by assumption we have T t β µ Ñ ν for a net pt β q.
Without loss of generality we can assume that T t β µ 0 converges to, say, the measure α and T t β µ s converges to, say, the measure ρ say. Then, we have By Proposition 3.13 we have that α P WAP 0 pGq and clearly ρ P SAPpGq. Thus, by uniqueness of the decomposition, we have α " 0. Hence, we have that ν " ν s " ρ belongs to the hull of µ s . This finishes the proof.
An immediate consequence is the following simple characterisation of null almost periodicity. To state it we will use the notation 0 to denote the zero measure on G.
Corollary 3.17. Let µ P WAPpGq. Then µ is null weakly almost periodic if and only if the zero measure 0 belongs to Xpµq. In this case δ 0 is the uniquely ergodic measure on Xpµq.
As is well-known, for weakly almost periodic systems unique ergodicity is equivalent to uniqueness of the minimal component. Indeed, each minimal component certainly carries ergodic measure and conversely uniqueness of such a component implies unique ergodicity [19,Prop. 2.10]. So from the previous corollary and Theorem 3.6 we directly infer the following result.
Remark. We note that unique ergodicity can also be derived from existence of means of weakly almost periodic functions: For any c 1 , .., c n P C c pGq and the function t Ñ ś n j"1 φ cj pT t µq is weakly almost periodic. Hence, its mean exists in a rather uniform manner. Since such linear combinations form a dense subset of CpXpµqq. one obtains uniform existence of means for all functions in CpXpµqq. This, in turn, implies unique ergodicity, see [35].
3.4. The structure of the hull. In this section we look a bit closer at the relation between Xpµq and Xpµ s q. This will allow us to get a rather precise description of the hull Xpµq.
The dynamical system pXpµ s q, Gq will play a central role in subsequent considerations. Hence, we will give it a special name.
As µ s is strongly almost periodic, we infer from Corollary 3.8 that Spµq is a compact abelian group. We denote the Haar measure on this group by θ Spµq . Moreover, by Theorem 3.15 we have that Spµq is a subset of Xpµq. Clearly, it is a closed G-invariant subset. Hence, from the unique ergodicity of pXpµq, Gq and the fact that θ Spµq is an invariant measure, we obtain the following corollary. Remark. Of course, this is just a special case of the general fact that a weakly almost periodic system with a unique minimal component has the Haar measure on this component as its unique translation invariant measure.
We next show that Spµq consists exactly of the strong almost periodic measures in Xpµq. First we present a more general result. Lemma 3.21. Let X be a weakly almost periodic dynamical system which has an unique minimal component S. Then, for an element ω P X we have ω P S if and only if Xpωq is an almost periodic dynamical system.
Proof. Assume ω P S. Since S is a minimal component, we have Xpωq " S. By the minimality of S we get that S is an almost periodic dynamical system.
Assume now that Xpωq is an almost periodic dynamical system. Hence, it must be minimal. Hence, by the uniqueness of the minimal component we get Xpωq " S and hence ω P S.
As a corollary we obtain the following.
Proof. Let ν P Xpµq. Then there exists some net pt α q in G such that Next, let us look at T tα µ s . As T tα µ s P Xpµ s q and Xpµ s q is compact, the net has cluster points. Thus, we can find some subnet pβq of pαq and some ω P Xpµ s q such that T t β µ s Ñ ω .
As T t β µ s P Xpµ s q we get ω P Xpµ s q. Therefore, by Proposition 3.11 we get that ω P SAPpGq. Now, as T t β µ 0 P Xpµ 0 q and Xpµ 0 q is compact, we can find a subnet γ of β such that T tγ µ 0 Ñ υ P Xpµ 0 q .
Then, by Proposition 3.13 we have υ P WAP 0 pGq. Now, we have The uniqueness of the decomposition gives us that υ " ν 0 . Since υ P Xpµ 0 q we get the claim. Now, we can prove that the projections P s and P 0 are continuous when restricted to the hull Xpµq.
Proof. First, let us observe that P s is well defined i.e. maps into Spµq. Indeed, if ν P Xpµq we know that ν P WAPpGq and hence P s pνq P SAPpGq X Xpνq Ă SAPpGq X Xpνq " Spµq .
In the same way P 0 is well defined.
We next prove the continuity of P s . To do this we have to show that whenever ν α Ñ ν in Xpµq we have pν α q s Ñ ν s .
As Xpµ s q " Spµq is compact, and pν α q s , ν s P Xpµ s q, the net pν α q s as well as any subnet has cluster points. To prove that pν α q s Ñ ν s , it thus suffices to show that every cluster point of pν α q s is equal to ν s .
Let ω be a cluster point of pν α q s . Then, there exists a subnet β of α such that ω " lim β pν β q s .
As pν β q s P SAPpGq X Xpµq " Spµq which is compact, hence complete in the vague topology, we get ω P Spµq and hence, ω P SAPpGq.
Moreover, by Lema 3.23, as ν β P Xpµq we have pν β q 0 P Xpµ 0 q. As before, by compactness of Xpµ 0 q we can also pick a subnet pν γ q 0 such that pν γ q 0 is convergent in Xpµ 0 q. Let υ be the limit of this net, then υ P WAP 0 pGq. Then As ω P SAPpGq and υ P WAP 0 pGq, the uniqueness of the decomposition shows that This shows that P s is continuous. Now, consider P 0 : Xpµq Ñ WAPpGq. Then, we have is the difference of two functions which are continuous on Xpµq, and hence P 0 is continuous on Xpµq. Restricting now the codomain to Xpµ 0 q will not change the continuity, and prove the claim.
Remark. For the preceding results the restriction to the hull of one element is crucial. On the whole space WAPpGq the maps P s and P 0 are not continuous with respect to the vague topology. Indeed, for example for G " R N it suffices to consider the sequence µ n " δ nZ N . Then, each µ n is of fully periodic and hence belongs to SAPpGq. However, the vague limit of the µ n is given by µ " δ 0 . So, in this case we then have P s pµ n q Ñ µ ‰ 0 " P s pµq and P 0 pµ n q " 0 Ñ 0 ‰ µ " P 0 pµq .
We summarize part of the preceding considerations as follows. S : Xpµq Ñ SpµqˆXpµ 0 q ; Spµq " pµ s , µ 0 q is a continuous G-mapping, one to one, and has full measure range.
Proof. The only thing which was not proven is pcq. The continuity of S is an immediate consequence of Theorem 3.24. The G-invariance of S follows immediately from the uniqueness of the Eberlein decomposition: T t µ " T t pµ s`µ0 q " T t pµ s q`T t pµ 0 q is the Eberlein decomposition of T t µ and therefore T t pµ s q " pT t µq s ; T t pµ 0 q " pT t µq 0 .
The fact that the S is one to one is obvious by the Eberlein decomposition. Finally, we have that has full measure in SpµqˆXpµ 0 q.
From these considerations we also easily infer the following result.
Proof. The mappings are well defined and continuous by Lemma 3.24. Now, if ν P Spµq then ν P SAPpGq. Therefore P s˝i pνq " P s pνq " ν s " ν .
Which proves the first relation. Also, for all ν P Xpµq we have i˝P s pνq " ν s .

Spectral and diffraction theory of weakly almost periodic measures
Based on the discussion in the preceding section we can rather directly set up the diffraction theory for a weakly almost periodic measures. In fact, the preceding section has immediate consequence for the spectral theory of the hull of the measure and this, in turn, gives results on the diffraction.
We start with the following rather direct consequence of Theorem 3.26. From this proposition we can infer that the spectral theory of the two dynamical systems pSpµq, Gq and pXpµq, Gq is the same. As the spectral theory of pSpµq, Gq is well-known we then obtain a precise description of the spectral theory of pXpµq, Gq.
To make this explicit we need a bit more notation.
Let pX, Gq be a dynamical system and µ an G-invariant probability measure on G. Then, an χ P p G is called an eigenvalue if there exists an f P L 2 pX, µq with f ‰ 0 and f˝T t " χptqf for all t P G. Such an f is then called an eigenfunction. A dynamical system is said to have pure point dynamical spectrum if there exists an orthonormal basis of L 2 pX, µq consisting of eigenfunctions.
Theorem 4.2. Let µ P WAPpGq. Then the dynamical system pXpµq, Gq has pure point dynamical spectrum with an orthonormal basis of continuous eigenfunctions given by Remark. (a) The group homomorphism G ÝÑ Spµq, t Þ Ñ T t µ, has dense range. Hence, its dual map j : z Spµq ÝÑ p G is injective and in this way the elements of z Spµq can be seen as elements of p G. A λ P z Spµq then gives rise to the eigenvalue jpλq. (b) The result implies that pSpµq, Gq is what is called the maximal equicontinuous factor of pXpµq, Gq. Indeed, one way to define this factor is as the dual to the group of all eigenvalues with continuous eigenfunctions equipped with the discrete topology, see e.g. [5] for a recent discussion in the context of diffraction. Now,in our case this clearly leads to Spµq.
(c) Uniquely ergodic systems with pure point spectrum with continuous eigenfunctions are called isomorphic extensions of their maximal equicontinuous factor. By the previous corollary the hull of a weakly almost periodic measure is an example of such a system. As shown recently by Downarowicz and Glasner [14] isomorphy of the extension is equivalent to mean equicontinuity in the minimal case.
(d) We have already met some eigenfunctions in Theorem 3.12. Let us emphasize that the eigenfunctions given in that theorem are completely canonical. This is rather remarkable as usually eigenfunctions are only determined up to some overall phase factor. Note also that Theorem 3.12 does not necessarily provide an eigenfunction for each eigenvalue (as the Fourier coefficients may vanish for some group elements).
Proof. As Spµq is a compact group, it is known to have pure point dynamical spectrum with an orthonormal basis of continuous eigenfunctions given by the elements λ P z Spµq. Now, the statement follows from the previous proposition and the continuity of P s .
As a consequence we also obtain the following application to diffraction theory. To express the corollary we will need the map j : z Spµq ÝÑ p G with jpλqptq :" λpT t µq (compare (a) of the preceding remark). A λ " |c jpλq pωq| 2 does not depend on ω P Xpµq and the diffraction is given by Remark. Validity of uniform convergence of the |c n λ | 2 to the point parts of p γ is sometimes discussed under the name of Bombieri-Taylor conjecture, see [31] for details. Thus, our result gives in particular validity of Bombieri/Taylor conjecture for the dynamical systems coming from weakly almost periodic measures.
Proof. We first show that the autocorrelations agree: As Xpµq, Xpµ s q are uniquely ergodic, each of µ and µ s have unique autocorrelation. Let us denote by γ 1 , γ 2 the two autocorrelations. Then, as Xpµq is uniquely ergodic, we have, see e.g. [7], c˚r c˚γ 1 ptq "ă φ c , T t φ c ą with the inner product calculated in L 2 pXpµq, θ µ q, where φ c is defined via φ c pνq " ν˚cp0q see above.
Then, as Xpµ s q is uniquely ergodic, we have c˚r c˚γ 2 ptq "ă φ c , T t φ c ą with the inner product calculated in L 2 pSpµq, θ µ q. By Theorem 3.20 those are equal, and hence c˚r c˚γ 1 " c˚r c˚γ 2 for all c P C c pGq. This shows that We now turn to proving pure point diffraction: This follows directly as the systems have pure point dynamical spectra. In fact, it is known that pure point diffraction and pure point dynamical spectrum are equivalent for translation bounded measure dynamical systems [7] (and in fact even more general situations [34,32]).
We finally discuss the formula for p γ: By the preceding corollary, the dynamical system pXpµq, Gq has pure point spectrum with continuous eigenfunctions to the eigenvalues jpλq, λ P Spµq. Thus, the uniform convergence of the c n λ to a function r c λ follows from Corollary 2 of [31]. Theorem 4 of [31] then gives that A λ :" |r c λ pωq| 2 is independent of ω P Xpµq and satisfies p γptjpλquq " A λ . Now, the equality p γ " ÿ λPSpµq A λ δ jpλq follows from Corollary 2 of [31]. It remains to show that r c λ pωq equals the Fourier coefficient c jpλq pωq. This, however, is clear from the definition of the Fourier coefficients.

On the hull of the autocorrelation
In the preceding considerations we essentially always started with a weakly almost periodic measure. Here, we discuss that any measure dynamical system gives rise to the hull of a weakly almost periodic measure viz its autocorrelation and how this hull carries important information on the pure point diffraction spectrum of the original system. In this sense hulls of weakly almost periodic measures are somehow unavoidable when dealing with diffraction.
Let pX, T q be a measure dynamical system and m an invariant probability measure on X. Then, there exists a unique positive definite measure γ with γ˚c˚r dp0q " xφ c , φ d y for all c, d P C c pGq, [7]. This measure is called the autocorrelation of m.
Collecting the results from this paper we get: Theorem 5.1. Let pX, G, mq be any ergodic system of translation bounded measures, and let γ be the autocorrelation of this system. Then (1) Xpγq is a compact group; (2) For m-almost all ω P X, Xpγq is isomorphic as compact group with Xpω s q. (3) Composing P s : X Ñ SAPpGq with the isomorphism from p2q gives us for m-almost all ω P X a continuous factor G-mapping Xpωq Ñ Xpγq.
Proof. (a) Since γ is positive definite and translation bounded, it is weakly almost periodic [40]. (b) Follows from (a) and Theorem 4.2.
(c) By Theorem 4.3, the intensity of the diffraction of pXpγq, Gq is given by It follows that A χ ‰ 0 ô p γptχuq ‰ 0, that is the systems pXpγq, Gq and pX, m, Gq have the same Bragg peaks. Now, the system pXpγq, Gq is weakly almost periodic, and hence it has pure point dynamical spectrum, which is the same as the group generated by the diffraction spectrum, [7].
(2) Since γ P SAPpGq the hull Xpγq is a compact group whose dual is exactly the pure point dynamical spectrum of Xpγq.
By [7] for almost all ω P X, γ is the autocorrelation of ω. Therefore, by (c), the diffraction and dynamical spectra of ω are exactly the dual of Xpγq.
As X Ă WAPpGq, it follows that for m-almost all ω, the systemXpωq has also pure point spectrum, and the spectrum is the dual of the compact group Xpω s q.
Therefore Xpω s q and Xpγq are compact groups with isomorphic duals.
Next, consider the set E of all dynamical systems of translation bounded measures on G equipped with invariant probability measures, that is E :" tpX, m, Gq|X Ă M 8 pGq, m invariant probability measure on Xu .
We can define a function F : where γ is the autocorrelation of pX, m, Gq and m 1 is the unique ergodic measure on Xpγq. Then, the previous theorem yields the following consequence. Remark. Note that while they are topologically isomorphic as dynamical systems, the systems F n ppX, m, Gqqq have typically different autocorrelation measures, and hence are in general supported on different subsets of M 8 pGq.
Finally, if we denote by γ p0q the autocorrelation of pX, m, Gq and by γ pnq the autocorrelation of F n pX, m, Gq we get (a) γ p0q P WAPpGq. (b) γ pnq P SAPpGq for all n ě 1. (c) For each n ě 1 and all χ P p G we have c χ pγ pn`1q q " |c χ pγ n q| 2 "ˇˇc χ pγ p0q qˇˇ2 n`1 .
In particular, all γ pnq have the same set of characters with non-trivial Fourier-Bohr coefficient.
Proof. (a) and (b) are immediate consequences of Proposition 5.2.
Since γ n`1 is Fourier transformable, we also have z γ n`1 ptχuq " c χ pγ n`1 q , which completes the proof.
Remark. The strong almost periodicity of γ 1 is the key for the proof of [47, Thm. 7.1].

Application to weighted Dirac combs
In this Section we apply part of the results of the previous sections to study the support of the Eberlein decomposition for weighted Dirac combs.
Recall that at subset Λ of G is called uniformly discrete if there exists an open set U containing the neutral element of G such that px`U q X py`U q " H for all x, y P Λ with x ‰ y. Such a subset is called relatively dense if there exists a compact set C such that Λ`C " G. A set which is both relatively dense and uniformly discrete is called a Delone set.
A set Λ Ă G is called a Meyer set if Λ is relatively dense and if Λ´Λ´Λ is uniformly discrete. If G is compactly generated, then the second condition can be replaced by the weaker Λ´Λ is uniformly discrete [9,47].
To connect subsets to the measures we note that any uniformly discrete set (and hence any Delone set) Λ gives rise to a measure where δ x denotes the unique point mass at x. This measure is known as the Dirac comb of Λ. The map δ : Uniform discrete subset of G ÝÑ M 8 is injective. In this way, the set of uniform discrete subsets of G inherits a topology and we can form also the hull XpΛq of Λ. Clearly, this hull is homeomorphic to Xpδ Λ q via δ.
6.1. On almost periodic Delone sets. Here, we consider the case that all weights are equal to one i.e. we look at some consequences of Theorem 3.15 for Delone sets.
Corollary 6.1. Let Λ be a Delone set such that δ Λ is a weakly almost periodic measure. Then there exists some Delone set Λ 1 so that pδ Λ q s " δ Λ 1 .
Remark. It doesn't follow from Proposition 6.1 that Λ itself is strongly almost periodic (in the sense that δ Λ is strongly almost periodic). For example, if Λ " Zzt0u then pδ Λ q s " δ Z which is indeed an element of Xpδ Λ q, but Λ is not strongly almost periodic. The problem is that while Z P XpΛq, we don't have Λ P XpZq.
An interesting question is what happens if we also ask for Λ to be minimal. We address this question next.
Let us recall first a Theorem of Favarov [20] and Kellendonk-Lenz [25]: Theorem 6.2. Let Λ be a Delone set with FLC. If δ Λ is strongly almost periodic, then Λ is a fully periodic crystal. Corollary 6.3. Let Λ be a Delone set with FLC. If δ Λ is weakly almost periodic and XpΛq is minimal, then Λ is a fully periodic crystal.
Combining Theorem 6.2 with Theorem 3.15 we get: Theorem 6.4. Let Λ be a Delone set with FLC such that δ Λ is weakly almost periodic. Then, there exists a lattice L and a finite set F so that F`L P XpΛq ; and pδ Λ q s " δ L`F " δ L˚δF .
Let us look next at two interesting examples of weakly almost periodic measures. Note that these examples are coming from Delone sets without FLC, and emphasize the importance of FLC in the previous results.
Let ν :" δ Λ´δZ . Then ν˚g is a function vanishing at infinity for all g P C c pRq [44], and thus null-weakly almost periodic [16]. This proves that ν is a null weakly almost periodic measure. Hence, using the uniqueness of the almost periodic decomposition, we have: pδ Λ q s " δ Z ; pδ Λ q 0 " ν .
A similar computation as in the previous example shows that, with Λ 1 " rZˆ2Zs Y rZ ?
Note that Λ 1 doesn't have FLC, and is the union of translates of lattices.
6.2. The support of the Eberlein decomposition. Given a weakly almost periodic measure µ, we know that µ s P Xpµq. We use this result to show that for weighted dirac combs the support of µ s and µ 0 cannot be much larger than the support of µ. We then use this result to rederive some recent results about Eberlein decomposition of weighted Dirac combs with Meyer set support [46].
Let us start by recalling a result of [4]. be a translation bounded measure supported inside S. If ν P Xpµq then there exists some S 1 P XpSq such that ν is supported inside S 1 .
Combining this result with Theorem 3.15 we get Theorem 6.8. Let µ be a weakly almost periodic measure supported on a set with FLC. Then µ s and µ 0 are pure point measures and suppµ s q has FLC.
Proof. It follows from Proposition 6.7 that suppµ s q has FLC. In particular µ s is a pure point measure. Since both µ and µ s are pure point measures, so is their difference µ 0 .
As consequence we get the following result about diffraction.
Theorem 6.9. Let µ be a translation bounded measure and let γ be an autocorrelation of ω. If γ is supported on an FLC set then the following statements hold: (a) p γ d P SAPp p Gq and p γ c P SAPp p Gq. (b) Each of the pure point and continuous spectra is either empty or relatively dense.
Proof. (a) By Theorem 6.8, each of µ s and µ 0 is a pure point measure. Moreover, since γ is positive definite, it is Fourier transformable, and hence so are µ s and µ 0 [40]. Now since the Fourier transform of a pure point Fourier transformable measure is a strongly almost periodic measure [28,40], (a) follows.
In particular we get new proofs of some of the results in [46].  We complete the section by showing that in this case, if we consider a cut and project scheme which gives our Meyer set, the decomposition doesn't go outside a translate of the closure of the window. This gives a result similar to [46,Thm. 4.6].
Let us start by recalling that a Cut and project scheme is a triple pG, H, Lq consisting of two LCAG's G respectively H and a lattice L Ă GˆH such that ‚ the restriction of the first projection π G : GˆH Ñ H to L is one to one. ‚ the second projection π H : GˆH Ñ H satisfies π H pLq is dense in H.
Given any cut and project scheme pG, H, Lq, each pre-compact set W Ă H defines an uniformly discrete set NpW q Ă G via NpW q :" tx P G|there exists y P W such that px, yq P Lu .
Next we need to recall the following lemma. be a translation bounded measure. If ν P Xpµq then there exists some ps, tq P GˆH such that suppνq Ă s`Npt`W q .
Combining this result with Theorem 3.15 we get (compare with [46]) Theorem 6.12. Let pG, H, Lq be a cut and project scheme, W Ă G be compact and µ :" ÿ xPNpW q ω x δ x , be a weakly almost periodic measure. Then there exists some ps, tq P GˆH such that suppµ s q Ă s`Npt`W q and suppµ 0 q Ă ps`Npt`W qq Y NpW q .
Remark. As the difference µ´µ s is a null weakly almost periodic measure with Meyer set support, it is essentially a small measure, and hence their supports cannot differ "too much". This is most likely the reason why in [46] the author proved that ps, tq in Theorem 6.12 can be chosen to be p0, 0q.

Eberlein convolution of weakly almost periodic measures
In this section we have a thorough look at the Eberlein convolution of weakly almost periodic measures measures. In particular, we show that this convolution always exists and is independent of the choice of van Hove sequence. We then go on and prove that it defines a strongly almost periodic measure. As an application we obtain an alternative proof for the pure pointedness of the autocorrelation of a weakly almost periodic measure (already shown above in Section 4). exists then for all f, g P C c pGq we have ̟˚f˚gptq " lim n 1 θ G pA n q ż An pµ˚f qpsqpν˚gqpt´sqds .
Let K 0 be any compact set containing˘supppf q and let K " K 0 Y pt`K 0 q. We claim that pf˚µqpsq1 An psqpg˚νqpt´sq´pf˚µ An qpsqpg˚ν An qpt´sq ‰ 0 ñ s P B K pA n q . This implies f˚µpsq ‰ f˚µ An psq or g˚νpt´sq ‰ g˚ν An pt´sq. Therefore, there exists some r such that f ps´rqp1´1 An prqq ‰ 0. Then r R A n and s´r P supppf q Ă K.
Thus s P pGzA n q`K. Since s P A n we get s P B K pA n q . Subcase 1.b: g˚νpt´sq ‰ g˚ν An pt´sq then ż G gpt´s´rqp1´1 An prqqdµprq ‰ 0 .
Therefore, there exists some r such that gpt´s´rqp1´1 An prqq ‰ 0. Then r R A n and t´s´r P supppf q Ă K 0 . Therefore s P t´r´K 0 Ă t´K 0´r Ă K´r .
As r R A n "´A n it follows that s P pGzA n q`K. This completes Case 1. Case 2: s R A n . Then, exactly as before pf˚µqpsq1 An psqpg˚νqpt´sq ‰ pf˚µ An qpsqpg˚ν An qpt´sq implies pf˚µqpsq1 An psq ‰ pf˚µ An qpsq or 1 An psqpg˚νqpt´sq ‰ pg˚ν An qpt´sq .
As 1 An psq " 0 we get 0 ‰ pf˚µ An qpsqpg˚ν An qpt´sq This implies that ş G f ps´tqdµ An ptq ‰ 0, and there fore, there exists some t P A n so that f ps´tq ‰ 0. Hence s´t P supppf q and therefore, s P A n`K .
Therefore, by the van Hove sequence property, we get that lim nˇş G pf˚µqpsqpg˚νqpt´sq1 An psqds θ G pA n q´ş G pf˚µ An qpsqpg˚ν An qpt´sqds θ G pA n qˇˇˇˇ" 0 .
In particular, in the vague topology, we have lim n pµ| An q˚Č pµ| An q´pν| An q˚Č pν| An q θ G pA n q " 0 .
Thus the limit lim n pµ|A n q˚Č pµ|A n q θGpAnq exists if and only if the limit lim n pν|A n q˚Č pν|A n q θGpAnq exists, and in this case they are the same.
Remark. In Lemma 7.10 the measures µ, ν need not be weakly almost periodic.