On average Hewitt–Stromberg measures of typical compact metric spaces

We study average Hewitt–Stromberg measures of typical compact metric spaces belonging to the Gromov–Hausdorff space (of all compact metric spaces) equipped with the Gromov–Hausdorff metric.

associated with the dimension function h; the lower and upper Hewitt-Stromberg measures of X associated with the dimension function h will be denoted by U h (X ) and V h (X ), respectively. The Hausdorff measure, the packing measure and the Hewitt-Stromberg measures satisfy the following string of inequalities (1. 1) in particular, the Hewitt-Stromberg measures form a natural interpolation between the Hausdorff measure and the packing measure. Hewitt-Stromberg measures were introduced by Hewitt and Stromberg in an intriguing exercise in their classical textbook from 1965 [10, (10.51)], and have subsequently been investigated further, see, for example [7,8,22], highlighting their importance in the study of local properties of fractals and products of fractals.
We now return to the main question: what are the fractal measures of a typical compact metric space? We are, of course, not the first to ask this question. Indeed, many different aspects of this problem have been studied by several authors during the past 20 years, including [1,4,6,12] and the references therein, and the question also appears implicitly in [5]. While (almost) all previous work, including, for example, [1,[4][5][6], study fractal measures of typical compact subsets of a given complete metric space, this paper adopts a new and different viewpoint introduced very recently by Rouyer [20] and investigated further in [12], namely, we investigate typical compact metric spaces belonging to the Gromov-Hausdorff space K GH of all compact metric spaces. For example, in [12] the authors prove the following result about the fractal measures of a typical compact metric spaces belonging to the Gromov-Hausdorff space K GH .
Theorem A [12] Let h be a continuous dimension function. A typical compact metric space We immediately note that the result in Theorem A is qualitatively similar to the behaviour of the Hausdorff measure of a typical compact subset of a fixed complete metric space. Indeed, it follows from [1,Remark 4.3] that if h is a right-continuous dimension function and X is a given fixed complete metric space, then a typical compact subset K of X satisfies H h (K ) = 0. We also note that Theorem A shows that the lower Hewitt-Stromberg measure (and hence the Hausdorff measure and the Hausdorff dimension) of a typical compact metric space is as small as possible and that the upper Hewitt-Stromberg measure (and hence the packing measure and the packing dimension) of a typical compact metric space is as big as possible. Other studies of typical compact sets, see [5,15,20], show the same dichotomy. For example, [20] proves that a typical compact metric space has lower box dimension equal to 0 and upper box dimension equal to ∞, and Gruber [5] and Myjak and Rudnicki [15] prove that if X is a metric space, then the lower box dimension of a typical compact subset of X is as small as possible and that the upper box dimension of a typical compact subset of X is (in many cases) as big as possible. The purpose of this paper is to analyse this intriguing dichotomy, and, in particular, the dichotomy in Theorem A, in more detail. We will prove that the behaviour of a typical compact metric space is spectacularly more irregular than suggested by Theorem A and the results in [5,15,20]. Namely, there are standard techniques, known as averaging systems, that (at least in some cases) can assign limiting values to divergent functions; the precise definition of an averaging system will be given in Sect. 2.5 below. This technique can be applied to the definition of the Hewitt-Stromberg measures as follows. Namely, for E ⊆ X and r > 0, let M r (E) denote the largest number of pairwise disjoint closed balls in X with centres in E and radii equal to r , and for a dimension function h, define the h-dimensional box counting function F h E (r ) of E by F h E (r ) = M r (E) h(2r ). The Hewitt-Stromberg measures U h (E) and V h (E) of a subset E of X are defined in terms of the lower and upper limits of the box counting function F h E (r ) of E, namely, (1. 3) The purpose of this paper is to show the following (surprising?) result: not only is the box counting function F h X (r ) of a typical compact metric space X so divergent that U h (X ) = 0 and V h (X ) = ∞, but it is so irregular that it remains spectacularly divergent even after being "averaged" or "smoothened out" by very general averaging systems (satisfying the mild closure-stability condition in Sect. 2.6). Specifically, if is an averaging system, then the associated average Hewitt-Stromberg measures satisfy U h (X ) = 0 and V h (X ) = ∞ for a typical compact metric space X ; more precisely, we prove the following theorem.
We present several applications of this result. For example, as an application of Theorem 1.1 we show that a typical compact metric space X is so irregular that the lower (upper) average Hewitt-Stromberg measures associated with all higher order Hölder averages of the box counting function F h X (r ) equal 0 (∞); below we state a precise version of this and refer the reader to Theorem 3.1 for a more general version of the result.
The higher order Hölder average Hewitt-Stromberg measures form a double infinite hierarchy between the Hausdorff measure and the packing measure in (at least) countably infinite many levels, namely, We emphasise that Theorems 1.1 and 1.2 are special cases of more general results presented in Sect. 2.
The paper is structured as follows. We first recall the definitions of the Gromov-Hausdorff space and the Gromov-Hausdorff metric in Sect. 2.1. In Sects. 2.2-2.3 we recall the definitions of the various fractal measures investigated in the paper. The definitions of the Hausdorff and packing measures are recalled in Sect. 2.2 and the definitions of the Hewitt-Stromberg measures are recalled in Sect. 2.3; while the definitions of the Hausdorff and packing measures are well-known, we have, nevertheless, decided to include these -there are two main reasons for this: firstly, to make it easier for the reader to compare and contrast the Hausdorff and packing measurers with the less well-known Hewitt-Stromberg measures, and secondly, to provide a motivation for the Hewitt-Stromberg measures. Section 2.4 recalls earlier results on the values of the Hausdorff measure, the packing measure and the Hewitt-Stromberg measures of typical compact metric spaces; this discussion is included in order to motivate our main results presented Sects. 2.5-2.6. In Sect. 2.5 we define average Hewitt-Stromberg measures, and in Sect. 2.6 we compute the exact values of average Hewitt-Stromberg measures of typical compact metric spaces. In Sect. 3 we apply the main results from Sects. 2.5-2.6 to the detailed study of average Hewitt-Stromberg measures associated with two of the most important types of averages, namely, higher order Hölder averages and higher order Cesaro averages. Finally, the proofs are given in Sects. 4-6. It is clear that ∼ is an equivalence relation in K GH , and the Gromov-Hausdorff space K GH is now defined as the space of equivalence classes, i.e.
While elements of K GH are equivalence classes of compact metric spaces, we will use the standard convention and identify an equivalence class with it representative, i.e. we will regard the elements of K GH as compact metric spaces and not as equivalence classes of compact metric spaces. Next, we define the Gromov-Hausdorff metric d GH on K GH . If Z is a metric space, and A and B are compact subsets of Z , then the Hausdorff distance d H (A, B) between A and B is defined by for X , Y ∈ K GH . The reader is referred to [18,Chapter 10] for a detailed discussion of the Gromov-Hausdorff space and the Gromov-Hausdorff metric. In particular, we note that it follows from [18] that the Gromov-Hausdorff space (K GH , d GH ) is complete (and, hence, not meagre) and the classification of subsets of K GH using Baire category is therefore meaningful.

Hausdorff measure and packing measure
While the definitions of the Hausdorff and packing measures (and the Hausdorff and packing dimensions) are well-known, we have, nevertheless, decided to briefly recall the definitions below. There are several reasons for this: firstly, since we are working in general (compact) metric spaces, the different definitions that appear in the literature may not all agree and for this reason it is useful to state precisely the definitions that we are using; secondly, and perhaps more importantly, the less well-known Hewitt-Stromberg measures (which will be defined below in Sect. 2.3) play an important part in this paper and to make it easier for the reader to compare and contrast the definitions of the Hewitt-Stromberg measures and the definitions of the Hausdorff and packing measures it is useful to recall the definitions of the latter measures; thirdly, in order to provide a motivation for the Hewitt-Stromberg measures. We start by recalling the definition of a dimension function. The Hausdorff measure associated with a dimension function h is defined as follows. Let X be a metric space and E ⊆ X . For δ > 0, we write The reader is referred to Rogers classical text for [19] for an excellent and systematic discussion the Hausdorff measures H h .
The packing measure associated with a dimension function h is defined as follows. For x ∈ X and r > 0, let C(x, r ) denote the closed ball in X with centre at x and radius equal to r , and for δ > 0, write is a family of pairwise disjoint closed balls in X with r i < δ and centres in E .

Hewitt-Stromberg measures
Hewitt-Stromberg measures were introduced by Hewitt and Stromberg in their classical textbook [10, (10.51)]. While Hausdorff and packing measures are defined using coverings and packings by families of sets with diameters less than a given positive number, δ say, the Hewitt-Stromberg measures are defined using packings of balls with the same diameter δ.
We will now recall the definition the Hewitt-Stromberg measures. Let X be a metric space and E ⊆ X . We first recall the definition of the packing number of E. For r > 0, the packing number M r (E) of E is defined by M r (X ) = sup |B| B is a family of pairwise disjoint closed balls in X with radii equal to r and centres in E . (2.1) Next, we recall the definition of the Hewitt-Stromberg measures associated with a dimension function h. For a metric space X and E ⊆ X , we define the lower and upper h-dimensional Hewitt-Stromberg pre-measures, denote by U h and V h , respectively, by Finally, we define the lower and upper h-dimensional Hewitt-Stromberg measures, denote by U h and V h , respectively, by The next result summarises the basic inequalities satisfied by the Hewitt-Stromberg measures, the Hausdorff measure and the packing measure.

Proposition 2.1 Let h be a dimension function. Then we have
for all metric spaces X and all E ⊆ X.

Hausdorff measures, packing measures and Hewitt-Stromberg measures of typical compact spaces
Jurina et al [12] have recently computed the Hausdorff measures, the packing measures and the Hewitt-Stromberg measures of typical compact spaces; this is the content of Theorem B below.
Theorem B [12] Hausdorff measures, packing measures and Hewitt-Stromberg measures of typical compact spaces. Let h be a continuous dimension function.
for all non-empty open subsets U of X . In particular, a typical compact metric space Theorem B shows that the lower Hewitt-Stromberg pre-measure and the lower Hewitt-Stromberg measure of a typical compact metric space are as small as possible and that the upper Hewitt-Stromberg pre-measure and the upper Hewitt-Stromberg measure of a typical compact metric space are as big as possible. We will analyse this intriguing dichotomy in more details by forming different types of averages. In order to do so, we introduce the following notation. Namely, for a dimensions function h and a subset E of a metric space X , 1 by introducing the following change of variables, namely, by letting r = e −t ; the reason for this change of variables is that it is more convenient to let t → ∞ when forming averages than letting r 0). Using the notation from (2.1), the Hewitt-Stromberg pre-measures of X are now given by and Theorem B therefore shows that the h-dimensional box counting function f h X (t) of a typical compact metric space X ∈ K GH diverges in the worst possible way as t → ∞. Below we analyse this divergence in detail using average procedures known as average systems.

Average Hewitt-Stromberg measures
We start by recalling the definition of an averaging (or summability) system; the reader is referred to Hardy's classical text [9] for a systematic treatment of averaging systems.

Definition (Average system) An averaging system is a family
is a positive measurable function, then we define lower and upper -average of f by respectively.
Applying averaging systems to the box counting function f h E (t) in (2.2) leads to the definition of average Hewitt-Stromberg measures.

Definition (Average Hewitt-Stromberg measures) Let h be a dimension function and let
= ( t ) t≥t 0 be an averaging system. For a metric space X and E ⊆ X , we define the lower and upper -average h-dimensional Hewitt-Stromberg pre-measures of E by We define the lower and upper -average h-dimensional Hewitt-Stromberg measures of E by respectively Note that Hewitt-Stromberg measures are, in fact, average Hewitt-Stromberg measures. Indeed, if X a metric space and we let denote the average system defined by = (δ t ) t≥1 (where δ t denotes the Dirac measure concentrated at t), then clearly for all subsets E of X . Below we list the basic inequalities satisfied by the average Hewitt-Stromberg measures, the Hausdorff measure and the packing measure.

Proposition 2.2 Let h be a dimension function and let be an averaging system. Then we have
for all metric spaces X and all E ⊆ X.

Proof
The statement is not difficult to prove and we have therefore decided to omit the proof.

Average Hewitt-Stromberg measures of typical compact spaces
In this section we present our main results. Many of our main results are valid for arbitrary averaging systems. However, in order to obtain the most optimal results we occasionally will have to assume that the averaging system satisfies a mild technical condition, namely, the h-closure-stability condition given in the following definition.
We immediately note that two important classes of averaging system are closure-stable. Namely, Proposition 2.3 (below) shows that the trivial averaging system = (δ t ) t≥1 is h-closure-stable for any dimension function, and Proposition 2.4 (below) shows that if = ( t ) t≥t 0 is an averaging system and each measure t has a continuous density with respect to Lebesgue measure, then is h-closure-stable for any continuous dimension function.

denotes the Dirac measure concentrated at t) is h-closure-stable for any dimension function h.
Proof This follows easily from the definitions and we have therefore decided to omit the proof.

Proposition 2.4 Let = ( t ) t≥t 0 be an averaging system and assume that for each t there is a measurable function
for all Borel sets B and such that if we write T t = sup supp π t , then Then is h-closure-stable for any continuous dimension function h.
The proof of Proposition 2.4 is given in Sect. 4. We can now state the main result in this paper, namely, Theorem 2.5 below. Theorem 2.5 provides the following (surprising?) extension of Theorem B: not only is the box counting function f h of a typical compact metric space X divergent as t → ∞, but it is so irregular that it remains spectacularly divergent as t → ∞ even after being "averaged" or "smoothened out" using very general averaging systems including, as will be shown in Sect. 3, all higher order Hölder and Cesaro averages.
Note that if we apply Theorem 2.5 to the trivial average system defined by = (δ t ) t≥1 , then it follows from (2.4) that the statement in Theorem 2.5 reduces to Theorem B.
We will now apply Theorem 2.5 to study average Hewitt-Stromberg measures obtained by considering higher order Hölder and Cesaro averages of the box counting function f h X (t) = M e −t (X ) h(2e −t ); this is the contents of Sect. 3 below.

Hölder and Cesaro averages of Hewitt-Strimborg measurers of a typical compact metric space
Two of the most commonly used averaging system are Hölder averages and Cesaro averages.
We will now define these average systems and apply them to the box counting function f h For a > 0 and a positive measurable For a positive integer n, we now define the lower and upper n'th order Hölder averages of f by , The Cesaro averages are defined as follows. First, we define I f : (a, ∞) → [0, ∞) by For a positive integer n, we now define the lower and upper n'th order Cesaro averages of f by , It is well-known that the Hölder and Cesaro averages satisfy the following inequalities, namely,  then see, for example, [9, pp. 110-111].
Using Hölder and Cesaro averages we can now introduce average Hölder and Cesaro Hewitt-Stromberg measures by applying the definitions of the Hölder and Cesaro averages to the function f h . This is the content of the next definition.
Definition (Average Hölder and Cesaro Hewitt-Stromberg measures) Let X be a metric space and n an integer with n ≥ 0. We define the lower and upper n'th order average Hölder Similarly, we define the lower and upper n'th order average Cesaro Hewitt-Stromberg mea- The higher order average Hölder and Cesaro Hewitt-Stromberg measures form a double infinite hierarchy between the Hausdorff measure and the packing measure in (at least) countably infinite many levels, namely, we have (using (3.1)) As an application of Theorem 2.5, we will now show that the behaviour of a typical compact metric space X ∈ K GH is so irregular that not even the hierarchies in (3.2) formed by taking Hölder and Cesaro averages of all orders are sufficiently powerful to "smoothen out" the behaviour of the box counting function f h

Proof of Proposition 2.4
We now turn towards the proof of Proposition 2.4. More precisely, the purpose of this section is threefold. Firstly, we recall a technical auxiliary result due to Gruber [5] and Rouyer [20] about the packing number (defined in (2.1)) and the covering number (defined below) of a metric space; this result plays an important role in Sects. 5, 6 and is stated in Lemma 4.1. Secondly, we prove an auxiliary results about the average h-dimensional Hewitt-Stromberg measures associated with h-closure-stable average systems; this result also plays an important role in Sects. 5, 6 and is proven in Lemma 4.2. Thirdly, and finally, we prove Proposition 2.4 showing that if = ( t ) t≥t 0 is an averaging system and each measure t has a continuous density with respect to Lebesgue measure, then is h-closure-stable for any continuous dimension function h.
We first recall the statement in Lemma 4.1 below; this is a well-known result due to Gruber [5] and Rouyer [20] and lists some useful continuity properties of the packing number and the covering number of a metric space. Recall, that if X is a metric space and E ⊆ X , then the packing number M r (E) of E is defined by M r (X ) = sup |B| B is a family of pairwise disjoint closed balls in X with radii equal to r and centres in E . (4.1) We also define the covering number N r (E) of E by N r (E) = inf |B| B is a family of closed balls in X with radii equal to r that covers E .

(4.2)
We can now state Lemma 4.1   and an open subset W of X such that C ∩ W = ∅ and C ∩ W ⊆ E i 0 . We therefore conclude

6), that if h is a dimension function, then an average system is called
Finally, using (4.3) and taking infimum over all countable families (2) The proof of this statement is identical to the proof of the statement in Part (1) and is therefore omitted.
Thirdly, and finally, we prove Proposition 2.4 from Sect. 2. We start with a small lemma. for all compact metric spaces X and all E ⊆ X.
Proof Write δ = 1 − u ∈ (0, 1). It follows from the definition of the packing number of pairwise disjoint closed balls C(x i , r ) in X with radii equal to r and centres x i ∈ E. Since x i ∈ E, there is a point y i ∈ E such that consists of pairwise disjoint balls, we therefore conclude that (C( is a family of pairwise disjoint closed balls in X with radii equal to (1 − δ)r and centres y i ∈ E. This clearly implies that M r ( E ) ≤ M (

Proof of Proposition 2.4 Let X be a metric space and E ⊆ X . It is clear that
We will now prove these inequalities. Let ε > 0. Fix t ≥ t 0 . Next we define three numbers ρ t , δ t and u t as follows.
π t (u) . Since K t is compact and D t is continuous, we conclude that D t is uniformly continuous, and we can therefore find a positive real Definition of δ t . Since h is continuous, and therefore uniformly continuous on compact subintervals of (0, ∞), we can find a positive real number Definition of u t . We can clearly choose a positive number 0 < u t < 1 such that . (4.6) After having defined the numbers ρ t , δ t and u t , we put v t = | log u t | and note that u t e −s = e −(s+v t ) for all real numbers s. Also, if s ∈ [t 0 , T t ], then we conclude from this and Lemma 4.3 that , and (4.7) therefore implies that

. It follows from this and (4.8) that
Write ε t = ε d t and note that it follows from (4.9) that Next, it follows from the definition of v t = | log u t | and (4.6) that v t (sup t 0 ≤s≤T t +1 f h E (s)) (sup t 0 ≤s≤T t π t (s)) ≤ ε, and we therefore conclude from (4.10) that (4.11) Also, note that for all u ∈ [t 0 + v t , T t ], we have (u, v t ) ∈ K t and v t = | log u t | ≤ ρ t (by (4.6)), and it therefore follows from (4.4) that D t (u, v t ) ≤ 1 + ε. We deduce from this and (4.11) that Since the consistency condition implies that ε t = ε d t → ε as t → ∞, we conclude from (4.12) and the definitions of

Proof of Theorem 2.5.(1)
The purpose of this section is to prove Theorem 2.5. (1). Recall, that if h is a dimension function and X ∈ K GH , then we define the function f h X : . For a dimension function h, an averaging system = ( t ) t≥t 0 , and t, c > 0, write Proof Write We must now prove that F is closed in K GH . In order to show this, we fix a sequence (X n ) n in F and X ∈ K GH with X n → X . We must now prove that X ∈ F, i.e. we must prove that f h X d t ≥ c. We prove this inequality below. For brevity define functions ϕ, ϕ n : We now prove the following three claims.

Claim 1
We have sup n ϕ n d t < ∞.

Proof of Claim 1
The measure t has compact support, and we can therefore find T 0 ≥ t 0 , such that supp t ⊆ [t 0 , T 0 ]. It follows from Lemma 4.1 that M e −T 0 is upper semi-continuous, and so lim sup n M e −T 0 (X n ) ≤ M e −T 0 (X ). In particular, this implies that there is a constant K such that M e −T 0 (X n ) ≤ K for all n. For positive integers n and s ∈ [t 0 , T 0 ] we therefore conclude that ϕ n (s) , it therefore follows that sup n ϕ n d t = Claim 2 We have c ≤ lim sup n ϕ n d t .
Proof of Claim 2 Since X n ∈ F, we conclude that c ≤ f h X n d t = ϕ n d t for all n, whence c ≤ lim sup n ϕ n d t . Claim 3 For all s ≥ t 0 , we have lim sup n ϕ n (s) ≤ ϕ(s), and so lim sup n ϕ n d t ≤ ϕ d t .

Proof of Claim 3
This follows from the fact that M r : K GH → R is upper semi-continuous for all r > 0 by Lemma 4.1. This completes the proof of Claim 3.
Finally, we deduce immediately from Claims 2 and 3 that c ≤ lim sup n ϕ n d t ≤ ϕ d t = f h X d t .

Proposition 5.2 Let h be a dimension function and let = ( t ) t≥t 0 be an averaging system.
(1) For c ∈ R + , write Then T c is co-meagre.
Then T is co-meagre.
Proof (1) It suffices to show that there is a countable family (G u ) u∈Q + of open and dense subsets G u of K GH with ∩ u∈Q + G u ⊆ T c . For u ∈ Q + , we define the set G u by recall that the set L h, t,c is defined in (5.1). Below we prove that the sets G u are open and dense subsets of K GH with ∩ u∈Q + G u ⊆ T c ; this is the contents of the three claims below.

Claim 1 G u is open in K GH .
Proof of Claim 1 This follows immediately from Lemma 5.1. This completes the proof of Claim 1.

Proof of Claim 2
We first prove that {X ∈ K GH | X is finite} ⊆ G u . Indeed, if X is a finite metric space, then it is clear that f h X (t) = M e −t (X ) h(2e −t ) → 0, and the consistency condition therefore implies that f h X d t → 0. It follows from this that there is a number t with t > u and f h X d t < c, whence X ∈ L h, t,c ⊆ G u . This proves that {X ∈ K GH | X is finite} ⊆ G u . Next, since {X ∈ K GH | X is finite} ⊆ G u and {X ∈ K GH | X is finite} is dense in K GH , we conclude that G u is dense in K GH . This completes the proof of Claim 2.

Proof of Claim 3
Let X ∈ ∩ u∈Q + G u . We must now show that U h (X ) ≤ c. Since X ∈ ∩ u∈Q + G u ⊆ ∩ n G n , we conclude that for each positive integer n, we can find a positive number t n with t n > n such that X ∈ L h t n ,c , whence f h X d t n < c. It follows immediately from this that U h (X ) = lim inf t f h X d t ≤ lim inf n f h X d t n ≤ c, and so X ∈ T c . This completes the proof of Claim 3.

Proof of Theorem 2.5.(2)-(3)
The purpose of this section is to prove Theorem 2.5.(2)-(3). We start by introducing the following notation. First, recall that for a positive real number r , the covering number N r (X ) of a metric space X is defined in (2.1). Next, for a dimension function h and a metric space X , define the function g h X : (0, ∞) → (0, ∞) by Finally, for a dimension function h, an averaging system = ( t ) t≥t 0 and r , t, c > 0, write and L h, r ,t,c = X ∈ K GH there is a positive integer N and such that

Proof
Write We must now prove that F is closed in K GH . In order to show this, we fix a sequence (X n ) n in F and X ∈ K GH with X n → X . We must now prove that X ∈ F, i.e. we must prove that g h X d t ≤ c. We prove this inequality below. For brevity define functions ϕ, ϕ n : [t 0 , ∞) → [0, ∞) by We now prove the following two claims.

Claim 1
We have lim inf n ϕ n d t ≤ c.
Proof of Claim 1 Since X n ∈ F, we conclude that ϕ n d t = g h X n d t ≤ c for all n, whence lim inf n ϕ n d t ≤ c. It follows immediately from this and Fatou's Lemma that lim inf n ϕ n d t ≤ lim inf n ϕ n d t ≤ c. This completes the proof of Claim 1.

Claim 2
For all s ≥ t 0 , we have ϕ(s) ≤ lim inf n ϕ n (s), and so ϕ d t ≤ lim inf n ϕ n d t .

Proof of Claim 3
This follows from the fact that N r : K GH → R is lower semi-continuous for all r > 0 by Lemma 4.1. This completes the proof of Claim 2.
Finally, we deduce from Claim 1 and Claim 2 that g h Proof Let X ∈ L h, r ,t,c and let d X denote the metric in X . We must now find ρ > 0 such that B(X , ρ) ⊆ L h r ,t,c . Since X ∈ L h, r ,t,c , we conclude that here is a positive integer N and such that Define : X → R by (x) = min i d X (x, x i ) and note that is continuous. Since X is compact, we conclude that there is a point x 0 ∈ X such that (x 0 ) = sup x∈X (x). For brevity write r 0 = (x 0 ) = sup x∈X (x), and note that since Also, since C i is compact and C i ⊆ B(x i , r ), we conclude that For brevity write Finally, since C i ∈ h, t i ,c and h, t i ,c is open (by Lemma 6.1), we conclude that there is a positive real number ρ i > 0 with It follows from (6.4) and (6.5) that ρ > 0. We will now prove that We therefore fix Y ∈ B(X , ρ) and proceed to show that Y ∈ L h, r ,t,c . Let d Y denote the metric in Y . Since d GH (X , Y ) < ρ, it follows that we may assume that there is a complete for all x , x ∈ X , and d Y (y , y ) = d Z (y , y ) for all y , y ∈ Y . Below we use the following notation allowing us to distinguish balls in Y and balls in Z . Namely, we will denote the open ball in Y with radius equal to δ and centre at y ∈ Y by B Y (y, δ), i.e. B Y (y, δ) = {y ∈ Y | d Y (y, y ) < δ}, and we will denote the open ball in Z with radius equal to δ and centre at z ∈ Z by B Z (z, δ) It is clear that Hence, to prove that Y ∈ L h, r ,t,c , it suffices to show that Y = ∪ i B Y (y i , r ), (6.8) The proofs of (6.8)-(6.10) are the contents of the three claims below.

Proof of Claim 1 It is clear that
In order to prove the reverse inclusion, we let r ). This completes the proof of Claim 1.

Proof of Claim 2
Since C i ⊆ B X (x i , r ), it follows from the definition of the numbers r i = inf{s | C i ⊆ B(x i s)} and d i = r − r i , that . (6.12) Finally, combining (6.11) and (6.12) shows that We can now prove that r ). This completes the proof of Claim 2.

Proof of Claim 3
It is clear that K i is a closed subset of Y and so K i ∈ K GH . We now prove that In particular, since x ∈ C i , this shows that dist(y, C i ) ≤ d Z (y, x) ≤ ρ, and so y ∈ K i . We deduce from this that dist(x, K i ) ≤ d Z (x, y) ≤ ρ. Finally, taking supremum over all x ∈ C i shows that sup x∈C i dist(x, K i ) ≤ ρ. This completes the proof of (6.14). Next, we prove that Indeed, let y ∈ K i . Since y ∈ K i , it follows from the definition of K i that there is x ∈ C i such that d Z (y, x) ≤ ρ, and so dist(y, C i ) ≤ d Z (y, x) ≤ ρ. Finally, taking supremum over all y ∈ K i shows that sup y∈D i dist(y, C i ) ≤ ρ. This completes the proof of (6.15). Combining (6.14) and (6.15), we immediately conclude that d H ( This completes the proof of Claim 3.
It follows immediately from Claims 1-3 that Y ∈ L h r ,t,c .

Proposition 6.3 Let h be a continuous dimension function and let
= ( t ) t≥t 0 be an averaging system.
(1) For c ∈ R + , write Then T c is co-meagre.
Then T is co-meagre.
for all metric spaces and all E ⊆ X, then S is comeagre.
Proof (1) It suffices to show that there is a countable family (G r ,u ) r ,u∈Q + of open and dense subsets G r ,u of K GH such that ∩ r ,u∈Q + G r ,u ⊆ T c . For r , u ∈ Q + , we define the set G r ,u by recall, that the set L h, r ,t,c is defined in (6.2)-(6.3). We now prove that the sets G r ,u are open and dense subsets of K GH such that ∩ r ,u∈Q + G r ,u ⊆ T c ; this is the contents of the three claims below.
Proof of Claim 1 This follows immediately from Proposition 6.2. This completes the proof of Claim 1.
Proof of Claim 2 Let X ∈ K GH and ρ > 0. Also, let d X denote the metric in X . We must now find Y ∈ G r ,u such that d GH (X , Y ) < ρ.
Firstly, since X is compact, we can find a finite subset E of X such that d H (X , E) < ρ 2 . Note that it is clear that we can find a constant k > 0 such that for all n ∈ N, all s > 0 and all δ > 0, we have (6.16) Next, choose real numbers T , t such that T > t > u, and for positive integers n, define ϕ n : (t 0 , ∞) → (0, ∞) by ϕ n (s) = e sn h(2e −s ).
Note that ϕ n ∞, and it therefore follows from the Monotone Convergence theorem that ϕ n d T ∞ d T = ∞. In particular, this implies that we can choose a positive integer N with Put δ 0 = min( ρ 2 , r ) and let C = {z ∈ R N | |z| ≤ δ 0 }. Finally, we define the space Y by Y = E × C and equip Y with the supremum metric d Y induced by d X and |·|, i.e. d Y ( (x , z ), (x , z ) ) = max( d X (x , x ), |z − z | ) for x , x ∈ E and z , z ∈ C. It is clear that Y is compact, and so Y ∈ K GH . Below we show that d GH (X , Y ) < ρ and Y ∈ G r ,u .
It is clear that f and g are isometries and we therefore for i = 1, . . . , M. In order to prove that Y ∈ G r ,u , it suffices the show that Y ∈ L h, r ,t,c , and in order to show that Y ∈ L h, r ,t,c , it clearly suffices to prove that However, it follows from (6.16) that N e −s (C) ≥ ke s N , and we therefore conclude from (6.21) that g h C i d t i ≥ k e s N h(2e −s ) d T (s) = k ϕ N d T . (6.22) Finally, combining (6.17) and (6.22), we now deduce that g h C i d t i ≥ k c k = c, whence C i ∈ h, t i ,c . This completes the proof of (6.20). It follows immediately from (6.18)-(6.20) that Y ∈ L h, r ,t,c ⊆ G r ,u . This completes the proof of Claim 2.

Proof of Claim 3
Let X ∈ ∩ r ,u∈Q + G r ,u . We must now show that if U is an open subset of X with U = ∅, then V h (U ) ≥ c. We therefore let U be an open subset of X with U = ∅, and proceed to show that V h (U ) ≥ c. Since U is non-empty and open there is x 0 ∈ U and r 0 > 0 with B(x 0 , r 0 ) ⊆ U . Next, since X ∈ ∩ r ,u∈Q + G r ,u ⊆ ∩ n G r 0 2 ,n , we conclude that for each positive integer n, we can find a positive real number t n with t n > n such that X ∈ L h, r 0 2 ,t n ,c .
In particular, this implies that there is a positive integer N n and x n,1 , . . . , x n,N n ∈ X , C n,1 , . . . , C n,N n ⊆ X , t n,1 , . . . , t n,N n > t n , such that X = ∪ i B x n,i , r 0 2 , C n,i ⊆ B x n,i , r 0 2 for all i, C n,i ∈ h, t n,i ,c for all i.
Since x 0 ∈ X = ∪ i B(x n,i , r 0 2 ), we can choose an index i n ∈ {1, . . . , N n } such that x 0 ∈ B(x n,i n , r 0 2 ), whence B(x n,i n , r 0 2 ) ⊆ B(x 0 , r 0 ), and so C n,i n ⊆ B(x n,i n , r 0 2 ) ⊆ B(x 0 , r 0 ) ⊆ U . Since it follows from Lemma 4.1 that f h U (s) ≥ g h U (s) for all s, we conclude from this combined with the fact that C n,i n ∈ h, t n,in ,c , that f h U d t n,in ≥ g h U d t n,in ≥ g h C n,in d t n,in ≥ c. Finally, since t n,i n > t n > n and so t n,i n → ∞, we deduce from the previous inequality that V h (U ) = lim sup t f h U d t ≥ lim sup n f h U d t n,in ≥ c. This completes the proof of Claim 3.
(2) This statement follows immediately from Part (1) since clearly T = ∩ c∈Q + T c .
(3) Using Part (2), it clearly suffices to prove that T ⊆ S. (6.23) We will now prove (6.23). Let X ∈ T . We must now show that X ∈ S, i.e. we must show that if U is an open subset of X with U = ∅, then V h (U ) = ∞. We therefore let U be an open subset of X with U = ∅, and proceed to show that V h (U ) = ∞. Since U is non-empty and open there is x ∈ U and r > 0 such that B X (x, r ) ⊆ U . In particular, if we write C = B(x, r 2 ), then C is compact and C ⊆ B(x, r ) ⊆ U . Next, we prove the following claim.

Claim 4 If
V is an open subset of X with V ∩ C = ∅, then V h (V ∩ C) = ∞.

Proof of Claim 4 Let
V be an open subset of X with V ∩ C = ∅. We must now show that V h (V ∩ C) = ∞. As V ∩ C = ∅, it is possible to choose y ∈ V ∩ C. Since y ∈ V and V is open, we can choose ε > 0 such that B(y, ε) ⊆ V . Next, since y ∈ C = B(x, r 2 ), we choose z ∈ B(x, r 2 ) such that z ∈ B(y, ε). Finally, since z ∈ B(x, r 2 ) ∩ B(y, ε), we can find δ > 0 with B(z, δ) ⊆ B(x, r 2 ) ∩ B(y, ε), whence B(z, δ) ⊆ B(x, r 2 ) ∩ B(y, ε) ⊆ C ∩ V , and so Finally, it follows immediately from Claim 4 and Lemma 4.2 that V h (C) = ∞, and since C ⊆ U , this implies that V h (U ) = ∞.
We can now prove Theorem 2.5.