Abstract almost periodicity for group actions on uniform topological spaces

Abstract We present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr- and Bochner-type almost periodicity, and similar conditions, and how the equivalence among such conditions relates to properties of the group action and the uniformity. We complete the paper by demonstrating how various examples considered earlier all fit in our framework.


Introduction
Almost periodicity plays an important role in many areas of mathematics, from differential equations to aperiodic order.Let us recall that t is a period for a system if translating by t the entire system we obtain an identical copy of the original system.We say that a system is fully periodic if the set of periods is relatively dense.
As introduced by Bohr [5], the idea behind almost periodicity is to replace the periods by almost periods.These are elements t such that after translation the system "almost" agrees (with respect to a topology) with the original system.An element is called almost periodic if the sets of almost periods are relatively dense.Bohr's original definition was for the uniform convergence topology on the space of uniformly continuous bounded functions on the real line, and in this case the definition is equivalent to the Bochner condition that the closed hull of the function is a compact space in this topology.Many of these ideas have been extended to other topologies on spaces of functions on various (or all) locally compact abelian groups by Stepanov [28], Weyl [32], Besicovitch [4], to measures [3,10,8,14] and even to distributions [30].
Almost periodicity plays a fundamental role in the area of Aperiodic Order due its connection to pure point diffraction.This connection already appears (more or less explicit) in the work of Meyer [18], Lagarias [11] and Solomyak [24,25].A first study giving an explicit equivalence statement between pure point spectrum and almost periodicity appears (in a specific situation) in [3] (see [8] as well for related work).This was then extended and further studied in the framework of dynamical systems in [10,9,19,15,29].On a fundamental level the equivalence was then established and studied in an framework free of dynamical systems in [14].It seems fair to say that by now the connection between pure point spectrum and almost periodicity, as well as the importance of almost periodicity to Aperiodic Order is well-established and thoroughly understood.
In many situations, the Bohr and Bochner definition of almost periodicity are equivalent (see for example [10,8,19,16] just to name a few).Moreover, the hull of an almost periodic function/measure is often a compact Abelian group [13,15,12,19].Since the proofs in these related but different situations are similar, it is natural to ask if there may be any unified theory of almost periodicity which shows the equivalence between Bohr and Bochner definition in a very general situation.It is our goal here to answer this question.
Let us describe here our general approach.Both the Bohr and Bochner definition for almost periodicity are in terms of properties of the orbit of an element under the translation action of the group, and can be defined for an arbitrary group action on a nice topological space.While in most situations studied in literature the topology is metrizable, this is neither the case in all situations nor necessary.Here, we deal with the more general case of group actions on uniform spaces, that is where the topology is defined by a uniformity.We should emphasize here that many of the results in the paper can probably be extended to group actions on arbitrary topological spaces.The uniformity structure seems to play an important role when studying Bochner type almost periodicity, as in this case pre-compactness is equivalent to total boundedness, and this is the reason why the set up is of uniform spaces.
Given the action α of a locally compact Abelian group (LCAG) G on a uniform space X, we study the connection between the following three properties of an element x ∈ X: • (Bohr type almost periodicity) For each entourage U ∈ U the set P U (x) := {t ∈ G : (x, α(t, x)) ∈ U } of almost periods is relatively dense.• (Bochner type almost periodicity) The orbit closure {α(t, x) : t ∈ G} is compact in X.
• (pseudo-Bochner type almost periodicity) For each entourage U ∈ U the set P U (x) is finitely relatively dense.
Under the extra natural assumption that the uniformity is G-invariant (see Definition 3.1 below), we show in Lemma 4.3 that pseudo-Bochner type almost periodicity is simply the total boundedness of the orbit {α(t, x) : t ∈ G}.Therefore, for a G-invariant uniformity, Bochner type almost periodicity implies pseudo-Bochner type almost periodicity, and the two concepts are equivalent if the uniformity is also complete (Proposition 4.4).It is immediate from the definition that pseudo-Bochner type almost periodicity implies Bohr type almost periodicity.
If the group action is equicontinuous (see Definition 3.4), we show in Proposition 4.6 that the converse also holds and hence Bohr and pseudo-Bochner type almost periodicity are equivalent.Combining the results, and using the fact that for G-invariant actions, equicontinuity on orbit closure is equivalent to continuity at 0 (see Lemma 3.8), we get in Theorem 4.7 that for continuous, G-invariant actions on uniform spaces, Bohr, Bochner and pseudo-Bochner type almost periodicity are equivalent.Moreover, in this case, the orbit closure becomes a compact Abelian group (Theorem 4.12).We complete the paper by looking at some particular examples in Section 6.

Uniform topologies
In this section, we discuss the necessary background on uniform topologies needed for our considerations.The material is well-known (see for example [6,Ch. 2] or [17]).Let us start with the definition of uniformity.Definition 2.1 (Uniformity).Let X be a set.A nonempty set U consisting of subsets of X × X is called a uniformity on X if it satisfies the following conditions.
The elements U ∈ U are called entourages.
Any uniformity U on X induces a topology τ U on X as follows: For x ∈ X and U ∈ U we define Then, the sets {U [x] : U ∈ U } define a basis at x for the topology τ U .Specifically, a subset O belongs to τ U if and only if for any x ∈ O there exists U ∈ U with U [x] ⊆ O.We say that a topology is uniform if it is the topology induced by a uniformity.
Any metrisable topology is uniform.Indeed, if (X, d) is a metric space, for each r > 0 we can define U r := {(x, y) ∈ X × X : d(x, y) < r} . Then, is a uniformity on X, which induces the topology defined by the metric d.As it is instructive and may help the reader to get some perspective on the definition of entourages we briefly sketch the proof that U d is a uniformity: As d(x, x) = 0 for all x, the system U d contains the diagonal.The system U d is upward closed by definition.The third point holds trivially.The fourth point is a consequence of the triangle inequality and the last point is a consequence of the symmetry of the metric.
Remark 2.2.In the preceding considerations we do not need d to be a metric.It suffices that d is a pseudometric, i.e. a symmetric map d : where the value ∞ is allowed.
Let U be a uniformity.Then, τ U is Hausdorff if and only if for all x, y ∈ X with x = y there exists some U ∈ U such that (x, y) / ∈ U .Equivalently, τ U is Hausdorff if and only if All the uniform topologies we consider are assumed to be Hausdorff.
We next discuss the completion of the topology induced by a uniformity.Let us start by recalling the definition of a filter.Definition 2.3.A filter on a set X is a family F ⊆ P(X) of subsets of X with the following properties: If U is a uniformity on X, a filter F is called Cauchy if for all U ∈ U there exists some A uniform space (X, U ) is called complete if every Cauchy filter is convergent.
Let us briefly discuss how to characterize uniform Hausdorff topologies which are given by a metric.First, we need the following definition.Definition 2.4 (Basis of an entourage).Let U be a uniformity on X.A set B ⊆ U is called a basis of entourages if for all U ∈ U there exists some W ∈ B such that W ⊆ U .
The following well-known result (see e.g.[31]) characterizes the metrizability of a uniformity.
Theorem 2.5 (Characterization metrizability of entourage).Let U be a Hausdorff uniformity on X.Then, there exist a metric d on X such that U = U d if and only if there exists a countable basis of entourages B ⊆ U .
At the end of this section, let us review the notion of total boundedness and its connection to compactness.
The importance of totally boundedness is given by the following result.
Remark 2.8 (Characterization compactness).From the theorem we may easily derive the following characterization of compactness in a uniform space.A subset A ⊆ X is compact if and only if A is totally bounded and complete (in the topology inherited from U ).
The topology of a compact set is always uniform.

Group actions on uniform topological spaces
In this section, we consider groups acting on uniform spaces.The material is certainly known.
Let (X, U ) be a uniform Hausdorff space, and let τ U be the topology induced by U on X.Let G be a group.Then, a map α : G × X → X is called a group action if it satisfies: • α(e, x) = x for all x ∈ X (where e is the neutral element of G). • α(s, α(t, x)) = α(st, x) for all x ∈ X and t, s ∈ G.
If α : G × X −→ X is a group action, the group G is said to act on X (via α).
Next, we review and introduce various definitions for the action α, which will play a central role in this paper.Note that we do not at first assume any continuity property of α.Indeed, we have not assumed that G carries a topology so far.
We start by introducing the notion of G-invariance for uniformities and (pseudo) Ginvariance for metrics on X. Definition 3.1.Let α be an action of G on X.
(a) A uniformity U on X is called G-invariant if for each U ∈ U there exists a U ′ ∈ U such that, for all t ∈ G, we have for all x, y ∈ X and t ∈ G, we have d(α(t, x), α(t, y)) = d(x, y) .
(c) A metric d on X is called pseudo G-invariant if for each ε > 0 there exists a δ > 0 such that, for all x, y ∈ X with d(x, y) < δ and all t ∈ G, we have It is obvious that any G-invariant metric is pseudo G-invariant.It is also clear from the definition in (a) that U ′ itself is contained in U (as we can take t to be the neutral element of G).This will be used tacitly subsequently.
Note that another way to phrase (c) of the preceding definition is that the family α t , t ∈ G, (with α t (x) = α(t, x)) is equicontinuous.By changing the metric to an equivalent metric, we can then make α t into a families of isometries.This is discussed next.Lemma 3.2 (Invariance and pseudo invariance).Let α be an action of G on a metric space (X, d), and let U d be the uniformity on X defined by d.Then, the following statements are equivalent.
It follows immediately from the definition that we have for all x, y ∈ X and t ∈ G. Now, d may not be a metric.Indeed, since X is not necessarily compact, d could be unbounded, i.e there could exist some x, y ∈ X such that d(x, y) = ∞.
To fix this issue, define We claim that this is a metric which has the desired properties.
Next, the definition of d gives that d(x, y) = d(y, x) for all x, y and hence d ′ (x, y) = d ′ (y, x).
Finally, let us prove the triangle inequality.Let x, y, z ∈ X.We split the problem into three cases: Case 1: Assume d ′ (x, z) ≥ 1.Then, one finds Case 2: Assume d ′ (y, z) ≥ 1.Then, we get This proves that d ′ is a metric.
This follows immediately from (1) and the definition of d ′ .• d and d ′ induce the same uniformity.
First note that we have Next, let ε > 0 be given.Assume without loss of generality that ε < 1.Since d is pseudo-invariant, there exists some δ > 0 such that d(α(t, y), α(t, z)) < ε for all y, z ∈ X with d(y, z) < δ and all t ∈ G.In particular, we have Next, consider a uniform Hausdorff space (X, U ), and let τ U be the topology defined by U .Let G be a locally compact Abelian group (LCAG).We then write the group operation as addition with +, and denote the neutral element of G as 0 and the inverse of t ∈ G by −t.
Let α be an action of G on X. Recall that α is called continuous if is a continuous mapping with respect to the topologies of G and (X, τ U ).We want to look at a stronger version of continuity for α.To do so, we start with the following lemma.Lemma 3.3.Let (X, U ) be a Hausdorff uniform space and α : G × X −→ X the continuous action of an LCAG.The following assertions are equivalent.
The family of function {α x } x∈X is uniformly equicontinuous (i.e. for every U ∈ U there exists an open set O ⊆ G containing 0 with (α x (s), Proof.(i) =⇒ (ii): This is obvious, since Then, by (iii), there exists an open set O ⊆ G containing 0 such that This gives (α(s, x), α(t, y)) Now, let s − t ∈ O.Then, for all x ∈ X, setting y = α(t, x) in (2) we get This shows that (α x (s), α x (t)) ∈ U holds for all x ∈ X and all s − t ∈ O, proving the equicontinuity of this family.
We can now define the notion of equicontinuous group actions.
Definition 3.4 (Equicontinuous group action).Let α be an action of G on the uniform Hausdorff space (X, U ).We say that α is equicontinuous if it satisfies one (and thus all) the equivalent conditions of Lemma 3.3.
Remark 3.5.We will discuss examples in Section 6 where the action is not continuous, or continuous but not equicontinuous.
Remark 3.6.The action α is continuous if, for all U ∈ U and x ∈ X, the set The action α is equicontinuous if, for all U ∈ U , the set is an open neighbourhood of 0 in G.
Definition 3.7 (Orbit and hull).Let α be an action of G on the uniform Hausdorff space (X, U ).For x ∈ X the orbit of x O(x) is defined as The hull of x (also called the orbit closure of x), denoted by H U x , is defined as the closure of the orbit in (X, τ U ), i.e.
Next, let us show that for G-invariant uniformities, the continuity of the action implies the equicontinuity on orbit closures.Lemma 3.8 (Equicontinuity on orbit closures).Let α be an action of G on the uniform Hausdorff space (X, U ). Assume that α is continuous and U is G-invariant.Then, for each x ∈ X, α is equicontinuous on H U x .
Proof.Let U ∈ U , and let V ∈ U be such that Clearly, V ′ ⊆ V holds (as we can take t = 0 on the left hand side).Since α is continuous, there exists an open set O ⊆ G containing 0 such that for all t ∈ O. Now, let t ∈ O and y, z ∈ H x be arbitrary such that (y, z) and hence, again by by (3) (α(t + s, x), α(s, x)) ∈ V .

Bohr and Bochner type almost periodicity
For this entire section we let a LCAG G be given and α is an action of G on a uniform Hausdorff space (X, U ).We discuss the standard definitions of almost periodicity in this context.
Let A ⊆ G be any set.Then, A is relatively dense if there exists a compact set K such that We say that A is finitely relatively dense if there exists a set finite F ⊆ G such that A+F = G.
Clearly, a finitely relatively dense set is relatively dense.
We can now introduce the following definitions.
Definition 4.1.For x ∈ X and U ∈ U define (a) The element x ∈ X s called Bohr-type almost periodic if for all U ∈ U the set (a) Let α be an action of G on a uniform Hausdorff space (X, U ) such that the uniformity is G-invariant.Then, for all U ∈ U there exists some V ∈ U such that, for all x ∈ X and t ∈ G, we have (b) Consider the space Bap 2,A (G) of Besicovitch almost periodic functions on a LCAG G (see [14] for details) together with the uniformity U given by the metric d(f, g) := f − g b,2,A , and the action α(t, f ) = τ t f (see [14]).Then, an element f ∈ Bap 2,A (G) is Bohr-type almost periodic if and only if f is a mean almost periodic function.
We will see below more examples, where Bohr/Bochner-type almost periodicity is equivalent with some other standard notion of almost periodicity.For this reason, to avoid confusion, we used the names Bohr-type and Bochner-type almost periodicity in Definition 4.1, instead of the simpler Bohr or Bochner almost periodicity.
Next, we will see that there is a connection between the total boundedness of the orbit of an element x and pseudo Bochner type almost periodicity.Lemma 4.3.Let α be an action of G on (X, U ) and let U be G-invariant.Then, an element x ∈ X is pseudo Bochner-type almost periodic if and only if the orbit O(x) is totally bounded.
Proof.We start with a preliminary observation used repeatedly in the proof.Let U ∈ U .Due to the G-invariance, there exists a V ∈ U with (α(s, x), α(s, y)) ∈ U for all (x, y) ∈ V and s ∈ G. Hence, (α(t, x), α(r, x)) ∈ V for some r, s ∈ G implies Without loss of generality we can assume O(X) ∩ V j = ∅, as otherwise we can erase U j from our list.Therefore, for each 1 ≤ j ≤ n there exists some t j ∈ G such that α(t j , x) ∈ V j .We claim (which gives finitely relative denseness of P U (x) and, hence, Bochner-type-almost periodicity of x).Indeed, let t ∈ G.Then, holds and, hence, there exists some 1 ≤ j ≤ n such that α(t, x) ∈ V j .Therefore, follows from the observation at the beginning of the proof.
=⇒: Let U ∈ U be given.Let W ∈ U be given with W ⊆ U and W • W −1 ⊆ U .Choose V ∈ U with (α(s, x), α(s, y)) ∈ W for all s ∈ G and (x, y) ∈ V .Since x is pseudo Bochner-type almost periodic, there exist t 1 , . . ., t n with For each 1 ≤ j ≤ n define U j = {y ∈ X : (y, α(t j , x)) ∈ W } .Now, let t ∈ G be arbitrary.Then, there exists some 1 ≤ j ≤ n with t ∈ t j + P V (x) and hence (α(t − t j , x), x) ∈ V .
By combining this result with Theorem 2.7, we obtain the following connection between Bochner-type almost periodicity and pseudo Bochner-type almost periodicity.In fact, it is easy to see that the equivalence between Bochner-type and pseudo Bochnertype almost periodicity is a matter of the completeness of U .By definition, any pseudo Bochner-type almost periodic point is Bohr-type almost periodic (as a finitely relatively dense set is clearly relatively dense).Next, we want to show that for equicontinuous group actions, Bohr-type almost periodicity implies pseudo Bochner-type almost periodicity.This will allow us show, in the case of equicontinuous group actions of G on a complete G-invariant uniform Hausdorff space (X, U ), the equivalence between Bohr-type, Bochner-type and pseudo Bochner-type almost periodicity.Proposition 4.6.Let G act on a uniform topological Hausdorff space (X, U ) such that the action is equicontinuous, and let x ∈ X.Then, x is Bohr-type almost periodic (i.e.P U (x) is relatively dense for any U ∈ U ) if and only if x is pseudo-Bochner-type almost periodic (i.e. for all U ∈ U the set P U (x) is finitely relatively dense).
Proof.⇐=: This follows immediately from the fact that every finite set in G is compact.=⇒: Let U ∈ U be arbitrary.Since the action is equicontinuous, there exists an open set O ⊆ G and V ∈ U such that Since x is Bohr-type almost periodic, there exists a compact set K such that Next, since K is compact, and O is open, there exists some finite set F such that We will show that Indeed, let t ∈ G.Then, there exists some We claim that s + u ∈ P U (x), which will complete the proof.Indeed, s ∈ P V (x) implies that (α(s, x), x) ∈ V Since u, 0 ∈ O and (α(s, x), x) ∈ V by (5) we have (α(u, α(s, x)), α(0, x)) = (α(u + s, x), x) ∈ U and hence s + u ∈ P U (x) as claimed.
The next result is a consequence of the previous proposition.Proof.Note that U induces a uniformity U x on H U x , which is complete and G-invariant.By Lemma 3.8, the action α is equicontinuous on (H U x , U x ).Moreover, it is easy to see that x is Bohr-type, Bochner-type or pseudo Bochner-type almost periodic in (X, U ), respectively, if and only if x is Bohr-type, Bochner-type or pseudo Bochner-type almost periodic in (H U x , U x ).The equivalence of (i) and (iii) now follows from Proposition 4.6 and the equivalence of (ii) and (iii) follows from Proposition 4.4 (applied to (H U x , U x )).We complete the section by discussing how -if the uniformity is G-invariant and complete -H U x has a natural Abelian group structure.
Proposition 4.8.Let G act on a uniform topological Hausdorff space (X, U ) such that the uniformity U is G-invariant and α is continuous.Let x ∈ X be given such that H U x is complete.Then, the following statements hold.
is a uniformly continuous group homomorphism, with dense range.
Proof.Note first that α is equicontinuous on H U x by Lemma 3.8.(a) We first show that ⊕ defined by α(t, x) ⊕ α(s, x) := α(s + t, x) is well defined on O(x), defines an Abelian group structure, and addition and inversion are continuous with respect to the topology induced by U .
To show that addition is well defined consider t, t ′ , s, s ′ ∈ G with α(t, x) = α(t ′ , x) and α(s, x) = α(s ′ , x).Then a direct computation gives This shows that ⊕ is well defined.By definition, O(x) is closed under ⊕.
Associativity of the addition on G yields that ⊕ is associative.Indeed, a direct computation gives Since G is Abelian, it is obvious that ⊕ is commutative.Moreover, for all t ∈ G, we have This shows that x = α(0, x) is the identity in O(x).Finally, for all t ∈ G, we have This shows that (O(x), ⊕) is an Abelian group.
We now show that (O(x), ⊕) becomes a topological group when equipped with the topology induced by U .Indeed, let U ∈ U be any entourage.Let W be such that W •W ⊆ U .Since α is G-invariant, there exists some V ∈ U such that, for all (y, z) ∈ V and t ∈ G, we have (α(t, y), α(t, z)) ∈ W .
Thus, we have the following group homomorphisms: Since the inclusion i is uniformly continuous, to complete the proof we need to show that F ′ is uniformly continuous.
Let U ∈ U be arbitrary.Since U is G-invariant, there exists some U ′ ∈ U such that Next, since α is continuous, it is equicontinuous on H U x by Lemma 3.8.Thus, there exists an open set O ⊆ G containing 0 and V ∈ U such that Now, let s, t ∈ G be so that s − t ∈ U .Then , by (9) we have (α(s − t, x), α(0, x)) ∈ U ′ and hence, by (8) we have This shows that F ′ is uniformly continuous, and completes the proof.
Remark 4.9.(a) Let us emphasize that we do not need compactness (i.e.almost periodicity of any form) of H U x to obtain the group structure on H U x To illustrate this we include the following example.Consider the translation action of R on the vector space C u (R) of uniformly continuous and bounded functions equipped with the supremum norm In the case that the hull is compact we can even say more.
Corollary 4.10.Let G act on a uniform topological Hausdorff space X such that the uniformity U is G-invariant and α is continuous.If x ∈ X is Bochner-type almost periodic, then H U x is a compact Abelian group with the addition ⊕ induced by α(t, x) ⊕ α(s, x) := α(s + t, x) and F : G −→ H U x , t → α(t, x), is a continuous group homomorphism with dense range.
Proof.Since x is Bochner-type almost periodic, H U x is compact.Hence, it is also complete.Now, the statement follows from Proposition refp1.
The previous corollary can be understood in terms of the Bohr-compactification G b of G.This is the (unique) compact group with the universal property that there exists a continuous group homomorphism ι b : G −→ G b such that, for any continuous group homomorphism ψ : G −→ H into a compact group H, there exists a unique mapping Ψ : G b −→ H with ψ = Ψ • ι (see for example [20,Prop. 4

.2.6] for details).
Corollary 4.11.Let G act on a uniform topological Hausdorff space X such that the uniformity U is G-invariant and α is continuous.If H U x is a compact Abelian group with the addition ⊕ induced by α(t, x) ⊕ α(s, x) := α(s + t, x) , then there exists a surjective continuous group homomorphism Ψ : holds for all t ∈ G.

Proof. Since H U
x is a compact Abelian group, by the universal property there exists a continuous group homomorphism Ψ : Since F has dense range, so does Ψ.Furthermore, since G b is compact, so is F (G b ).It follows that the range of F is compact and hence closed.In particular, F is surjective.
Combination of the previous results gives our next main result.Theorem 4.12 (Main result -II).Let G act on a complete uniform Hausdorff topological X space such that the action is continuous and the uniformity U is G-invariant.Then, for x ∈ X, the following assertions are equivalent.
x is a compact Abelian group with the group operation ⊕ induced by α(t, x) ⊕ α(s, x) := α(s + t, x) .
(v) There exists a continuous function Ψ : Moreover, in this case F (t) = α(t, x) define a group homomorphism with dense range x , and Ψ is an onto group homomorphism.Proof.The equivalences between (i), (ii) and (iii) follow from Theorem 4.7.The implication (ii) =⇒ (iv) follows from Corollary 4.10.This Corollary gives also that F : G −→ H U x , t → α(t, x), is a continuous group homomorphism with dense range.The implication (iv) =⇒ (v) follows from Corollary 4.11.Next, we will show (v)=⇒(ii): Since Ψ is continuous, and G b is compact, so is Ψ(G b ) =: K.Moreover, by (v) we also have α(t, x) = Ψ(i b (t)) for all t ∈ G, which implies O(x) ⊂ K.As the compact K is closed we infer x must be compact as it is a closed subset of a compact set.This shows (ii).Moreover, as K is contained in H U x (by definition of Ψ) we have even x giving H U x = K and hence Ψ is onto.This completes the proof.

The mixed uniformity
In the preceding discussion and most notably in Lemma 3.8 we have seen that the continuity of α is equivalent to the equicontinuity of α on orbit closures (provided U is G-invariant).This -so to say -booster of continuity has been a main tool in our considerations.Now, it may happen that α is not continuous.However, even if we lack continuity, by a simple mixing process we can make α equicontinuous, without changing the Bohr-type almost periodicity, or the other basic properties of the uniformity.This is discussed in this section.
Note, however, that since the equivalence between Bohr-type and pseudo Bochner-type almost periodicity relies on equicontinuity, this mixing process can actually change pseudo-Bochner and Bochner type almost periodicity, and we will see such an example in the next section.
Let G be a LCAG and let U be a G-invariant uniformity on some set X and α : G× X → X any group action.Denote by O the set of all open sets in G containing 0. Now, for U ∈ U and O ∈ O define and set Let us first show that U mix is indeed a uniformity on X.
Proposition 5.1 (U mix as uniformity).If U is G-invariant, then U mix is a uniformity on X.
Proof.We have to show that the five points defining a uniformity are satisfied.It is rather straightforward to show this.For the convenience of the reader we include a proof.
• Let V ∈ U mix .Then, there exists some O ∈ O and U ∈ U so that V [O, U ] ⊆ V .Then, for all x ∈ X we have 0 ∈ O and (x, x) ∈ U and hence This shows that ∆ ⊆ V .
Moreover, it follows immediately from the definition that for 1 ≤ j ≤ 2 we have By the G-invariance of U , there exists some Note that U ′ is contained in U (as we can set t = 0) and set Then, there exists some z ∈ X such that (x, z), (z, y) Note here that z ′ = α(−s, α(s, z ′ )) = α(−s, z) and x ′ = α(−t, x) hold.Therefore, by (10) we have (α Hence, we obtain Definition 5.2 (Mixed uniformity).Let the LCAG G act on the uniform space (X, U ) such that U is a G-invariant.Then, U mix is called the mixed uniformity induced from the action of G. Proposition 5.3 (Characterization of U mix ).Let the LCAG G act on the uniform space (X, U ) such that U is G-invariant.Then, the following statements hold.
(a) The action α is equicontinuous on (X, U mix ) and U is finer than U mix , that is U mix ⊆ U holds.(b) U is the finest uniformity satisfying (a), i.e. whenever U ′ is a uniformity such that U is finer than U ′ and α is equicontinuous on (X, U ′ ), then U ′ ⊆ U mix holds.
In particular α is equicontinuous on (X, U ) if and only if U = U mix .
Proof.The last statement is immediate from (a) and (b).
(a) From Lemma 3.3 (iii) it follows easily that α is equicontinuous with respect to (X, U mix ).
To show that U is finer that U mix we consider an arbitrary V ∈ U mix .Then, there exists some ] holds and we get U ⊆ V .Since U is a uniformity, we get V ∈ U and therefore U mix ⊆ U .follows as V ∈ U mix was arbitrary.
(b) Let U ′ with U ′ ⊆ U be given such that α is equicontinuous on (X, U ′ ).Let V ∈ U ′ be arbitrary.Since α is equicontinuous on (X, U ′ ), by Lemma 3.3 (ii), there exists some Now, let t ∈ P V (x) be arbitrary.This means and therefore, there exists some s ∈ O and y ∈ X such that α(t, x) = α(s, y) and (y, x) ∈ U holds.The first relation gives y = α(t − s, x) and therefore, (α(t − s, x), x) ∈ U .
Thus we obtain Since t ∈ P V (x) was arbitrary, we get and hence Since O + K is compact, we get that P U (x) is relatively dense.As U ∈ U was arbitrary, the claim follows.
When we now collect all results from this section and combine them with Theorem 4.12, the following result is an immediate consequence.
Corollary 5.8.Let G act on a complete metric space (X, d) space such that d is pseudo Ginvariant, and let U = U d .Then, for x ∈ X, the following assertions are equivalent.
(vi) There exists a continuous function Ψ : Moreover, in this case F (t) = α(t, x) define a group homomorphism with dense range F : G → H U x , and Ψ is an onto group homomorphism.

Examples
In this section we present a wealth of examples for our results.This will in particular how all earlier corresponding results on almost periodicity that we are aware of fall within our framework.
6.1.Bohr and Bochner almost periodicity for functions.Consider X to be the vector space C u (G) of uniformly continuous bounded functions on a LCAG G with the topology given by • ∞ and the translation action It is easy to see that α is G-invariant, equicontinuous and that the uniformity is complete.Therefore, Thm.4.12 gives the following well-known result.
(i) f is Bohr-almost periodic.
(iii) H f := {T t f : t ∈ G} is a compact Abelian group with the addition operation induced by (iv) There exists a continuous function Ψ : Moreover, in this case F (t) = α(t, x) define a group homomorphism with dense range F : G → H f , and Ψ is an onto group homomorphism.
Note here that the mapping δ 0 : is uniformly continuous, and hence so is δ 0 • Ψ ∈ C(G b ).Then, (iv) in Theorem 6.1 implies that there exists some g It follows easily from here that (iv) in Theorem 6.1 can be replaced by (iv') There exists some g ∈ C(G b ) such that f = g • i b .
6.2.Group valued almost periodic functions.In this section we review the concept of group valued almost periodic functions, as discussed recently in [12].Let G, H be two LCAG, with H complete.Let H be the set of all functions f : G → H.For each neighbourhood W ⊆ H of 0 define

6.4.
Stepanov almost periodicity.Here we follow closely [26].Let G be a LCAG and K ⊆ G be a fixed compact set with non-empty interior, and let 1 ≤ p < ∞.
Let BS p K (G) denote the space of all f ∈ L p loc (G) such that Then, (BS p K (G), Next, let A be an exhaustive nested van Hove sequence (see [19] for definitions and properties).Let U denote the uniformity defined by d.Note here that the uniform discreteness of elements in D U imply that the translation action is never continuous.Since (D, U ) is a complete uniform space, U is G-invariant and both the topology of G and U are metrisable, we can mix the uniformity as in Section 5, and then [19,Lem. 4.5 and Prop. 4.6] are simply consequences of Corollary 5.8.6.7.Vague topology.Consider now X = M(G) the space of measures on G equipped with the vague topology, and the translation action of G. Since the vague topology is complete, Bochner-type and pseudo-Bochner type almost periodicity are equivalent.Moreover, a measure µ ∈ X is Bochner-type almost periodic if and only if it is translation bounded [2,27].Now, the translation action of G on X is not equicontinuous, so Bochner and Bohr type almost periodicity may not be equivalent.Since every translation bounded measure is Bochnertype almost periodic, it is automatically Bohr-type almost periodic.
Next, consider the measure µ := Then, by construction µ N is 5 N periodic and supp(ν N ) ⊆ 2 • 5 N + 5 N +1 Z.In particular, for all m ∈ 5 N + 5 N +1 Z, we have This immediately implies that µ is Bohr type almost periodic for the vague topology.Indeed, let U be an entourage for the vague topology.Then, there exists some ϕ 1 , . . ., ϕ n ∈ C c (G) and ǫ > 0 such that Let N be so that, for all 1 ≤ j ≤ N we have supp(ϕ j ) ⊆ (−5 N , 5 N ) .
Let m ∈ 5 N + 5 N +1 Z be arbitrary.By (11), we have This shows that µ is Bohr-type almost periodic, but, since it is not translation bounded, it is not Bochner-type almost periodic.
6.8.Product topology.Let G act via translation of functions on a uniform space of (X, U ) of functions on G, with the property that C u (G) ⊆ X.We can then push the uniformity U to M ∞ (G) the following way.
For each U ∈ U , n ∈ N and all ϕ 1 , . . ., ϕ n ∈ C c (G) define Next, define Then, U pu is a uniformity on M ∞ (G).If U is G-invariant,so is U pu .Moreover, if the translation action is equi-continuous on U , it is also equi-continuous on U pu .Completeness is in general more subtle.This process allows us carry many results about almost periodicity from functions to measures.We look next at one such example.Now, consider the case when X = C u (G) with the uniformity given by the norm • ∞ .The uniformity induced by the push of this uniformity to M ∞ (G) is called the product uniformity, and is denoted by U p .The corresponding topology is the product topology for measures (see [8] for properties).It is easy to see that U p is G-invariant, and the translation action is equicontinuous.Now, it is not known if this space, or if M ∞ (G) is complete, but this space is quasi-complete (meaning any equi-translation bounded closed subset is complete [8]).This turns out to be enough in this situation.Indeed, if µ ∈ M ∞ (G), then all measures in H Up µ are equi-translation bounded, and hence this orbit is complete.Therefore, we get the standard characterization of SAP(G), the first three equivalences appearing in [8,20], while the equivalence to condition (iv) appearing in [13,16,12].Theorem 6.4.Let µ ∈ M ∞ (G).Then, the following assertions are equivalent.Moreover, in this case F (t) = α(t, x) define a group homomorphism with dense range F : G → H Up µ , and Ψ is an onto group homomorphism.6.9.Norm and mixed norm topology.Consider X = M ∞ (G), the space of translation bounded measures.Let K ⊆ G be a fixed compact set with non-empty interior.Then, K defines a norm on X [3] via µ K := sup t∈G |µ| (t + K) .
The space (X, • K ) is a Banach space [22].Let U norm be the uniformity defined on M ∞ (G) by this norm.
Let us first recall the following definition [3].Norm almost periodic pure point measures are appear naturally in the CPS [3,21,13, 29] and have been fully characterized in [29].
Note that the translation action is not continuous on X, and while it is continuous on some orbits (for example it is continuous on the orbit of measures of the form µ = f θ G for f ∈ C u (G)) it is never continuous on the orbit of non-trivial pure point measures (see Lemma 6.6 below).In particular, in this case we have Bochner-type almost periodicity ⇐⇒ pseudo Bochner-type almost periodicity =⇒ Bohr-type almost periodicity .
It turns out that for pure point measures, Bochner-type almost periodicity is not a relevant concept, as we have the following lemma.This contradicts the fact that T is continuous at s. Next, fix some open precompact set U with 0 ∈ U and set ε = a 2 .We show that P ε := {t ∈ G : T t µ − µ K < ε} is locally finite and hence countable, by the σ-compactness of G. Since P ε is finitely relatively dense, this implies that G is countable, a contradiction.Indeed, let K ⊆ G be any compact set and let We want to show that F is finite.giving that F is finite, and completing the proof.
Since U norm is given by a metric, and G-invariant, we can define the mixed uniformity from Section 5. We will denote the mixed uniformity by U m−n and refer to the topology induced by this uniformity as the mixed-norm topology for measures.Now, Corollary 5.8 has the following consequence.
(ii) H  , and Ψ is an onto group homomorphism.

Proposition 4 . 4 .
Let G act on a uniform topological Hausdorff space and let U be G-invariant.Then, the following holds for x ∈ X: (a) If x is Bochner-type almost periodic then x is pseudo Bochner-type almost periodic.(b) If (X, τ U ) is complete and x is pseudo Bochner-type almost periodic then x is Bochnertype almost periodic.

Corollary 4 . 5 .
Let G act on a uniform topological Hausdorff space and assume that the uniformity is G-invariant.Let x ∈ X be pseudo-Bochner almost periodic.Then x is Bochner almost periodic if and only if H U x is complete.

Theorem 4 . 7 (
Main result -I).Let G act on a complete uniform topological Hausdorff space (X, U ) such that the action is continuous and U is G-invariant.Let x ∈ X.Then, the following statements are equivalent.(i)x is Bohr-type almost periodic.(ii) x is Bochner-type almost periodic.(iii) x is pseudo Bochner-type almost periodic.
and let U = {U ⊆ H × H : there exists W such that U W ⊆ U } .As usual, let C(G : H), C b (G : H) and C u (G : H) denote the subspaces of H consisting of continuous, continuous bounded, and uniformly continuous and bounded functions, respectively.From [12, Lem.6] we find the following.Lemma 6.2.(a) U is a uniformity on H.(b) H is complete with respect to U .(c) C(G : H), C b (G : H) and C u (G : H) are closed in H.

( v )
There exists a continuous function Ψ :G b → H Up µ such that T t f = Ψ(i b (t)) for all t ∈ G .

Definition 6 . 5 .
A measure µ in M ∞ (G) is called norm almost periodic if µ is Bohr-type almost periodic in this topology.

( b )
Assume by contradiction that µ = 0.The argument is similar to the one in (a).Since µ is pure point, there exists some x ∈ G such that |µ({x})| = a > 0 .

( v )
There exists a continuous function Ψ :G b → H U m−n µ such that T t f = Ψ(i b (t)) for all t ∈ G .Moreover, in this case F (t) = α(t, x) define a group homomorphism with dense range F : G → H U m−n µ • S p K ) is a Banach space [26, Prop.2.2].Now, let U step be the uniformity defined by this norm on BS p K (G).Since the Stepanov norm is G-invariant by definition, so is U step .Moreover, the translation action is continuous [26, Lem.2.5] and hence uniformly continuous on every H Bohr-type almost periodicity with respect to this uniformity is called Stepanov almost periodicity.Therefore, Stepanov almost periodicity is equivalent to Bohr-type almost periodicity with respect to this uniformity, giving [26, Prop.2.7].6.5.Weak almost periodicity.Consider X = C u (G) with the weak topology of the Banach space (C u (G), • ∞ ).Note here that the weak topology is not complete for infinite dimensional vector spaces, and hence is not complete on C u (G) unless G is a finite group.Moreover, the translation action T t f (x) = f (x − t) is not equicontinuous.Because of this, in this case we only get the implications Bochner-type almost periodicity =⇒ pseudo Bochner-type almost periodicity =⇒ Bohr-type almost periodicity .
6.6.Autocorrelation topology.Let G be a second countable LCAG, and let A = (A n ) be a van Hove sequence (which exists due to the second countability of G, see [27, Prop.B.6]).Fix an open set U = −U ⊆ G and define