# On average Hewitt–Stromberg measures of typical compact metric spaces

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## Abstract

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.

## Keywords

Box dimension Compact metric space The Gromov–Hausdorff metric Hewitt–Stromberg measures Hausdorff measure Packing measure## Mathematics Subject Classification

28A78 28A80## 1 Introduction

Recall that a subset *E* of a metric space *M* is called co-meagre if its complement is meagre; also recall that if \(\textsf {P}\) is a property that the elements of *M* may have, then we say that a typical element *x* in *M* has property \(\textsf {P}\) if the set \(E=\{ x \in M \, |\,x\text { has property } \textsf {P}\}\) is co-meagre, see Oxtoby [16] for more details. The purpose of this paper is to investigate the average Hewitt–Stromberg measures of a typical compact metric space belonging to the Gromov–Hausdorff space \(K_{\textsf {GH}}\) of all compact metric spaces; the precise definitions of the average Hewitt–Stromberg measures and the Gromov–Hausdorff space \(K_{\textsf {GH}}\) will be given below.

*X*are the

*h*-dimensional Hausdorff measure \({\mathcal {H}}^{h}(X)\) and the

*h*-dimensional packing measure \({\mathcal {P}}^{h}(X)\) associated with the dimension function

*h*; the precise definitions of \({\mathcal {H}}^{h}(X)\) and \({\mathcal {P}}^{h}(X)\) will be given in Sect. 2.2. It is well-known that these measures satisfy the following inequality,

*X*, namely, the lower and upper Hewitt–Stromberg measures 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 \({\mathcal {U}}^{h}(X)\) and \({\mathcal {V}}^{h}(X)\), respectively. The Hausdorff measure, the packing measure and the Hewitt–Stromberg measures satisfy the following string of inequalities

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 \(K_{\textsf {GH}}\) 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 \(K_{\textsf {GH}}\).

### Theorem A

*h*be a continuous dimension function. A typical compact metric space \(X\in K_{\textsf {GH}}\) satisfies

*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 \({\mathcal {H}}^{h}(K)=0\). 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 \(\infty \), 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 \(E\subseteq X\) 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 \(F_{E}^{h}(r)\) of

*E*by

*E*of

*X*are defined in terms of the lower and upper limits of the box counting function \(F_{E}^{h}(r)\) of

*E*, namely,

*X*so divergent that \({{\mathcal {U}}}^{h}(X)=0\) and \({{\mathcal {V}}}^{h}(X)=\infty \), but it is so irregular that it remains spectacularly divergent even after being “averaged” or “smoothened out” by very general averaging systems \(\Pi \) (satisfying the mild closure-stability condition in Sect. 2.6). Specifically, if \(\Pi \) is an averaging system, then the associated average Hewitt–Stromberg measures satisfy \({{\mathcal {U}}}_{\Pi }^{h}(X)=0\) and \({{\mathcal {V}}}_{\Pi }^{h}(X)=\infty \) for a typical compact metric space

*X*; more precisely, we prove the following theorem.

### Theorem 1.1

*h*be a continuous dimension function and let \(\Pi \) be an averaging system satisfying the

*h*-closure stability condition in Sect. 2.6. A typical compact metric space \(X\in K_{\textsf {GH}}\) satisfies

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 \(F_{X}^{h}(r)\) equal 0 (\(\infty \)); below we state a precise version of this and refer the reader to Theorem 3.1 for a more general version of the result.

### Theorem 1.2

*h*be a continuous dimension function and \(n\in \mathbb N\cup \{0\}\). We define the

*n*’th order Hölder averages, denoted by \(F_{E,n}^{h}(t)\), of the box counting function \(F_{E}^{h}(r)\) of a subset

*E*of a metric space

*X*inductively by

*n*’th order Hölder average Hewitt–Stromberg measures of

*X*by

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.

## 2 Statements of results

### 2.1 The Gromov–Hausdorff space \(K_{\textsf {GH}}\) and the Gromov–Hausdorff metric \(d_{\textsf {GH}}\)

*Z*is a metric space, and

*A*and

*B*are compact subsets of

*Z*, then the Hausdorff distance \(d_{\textsf {H}}(A,B)\) between

*A*and

*B*is defined by

### 2.2 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.

### Definition

(*Dimension function*) A function \(h:(0,\infty )\rightarrow (0,\infty )\) is called a dimension function if *h* is increasing, right continuous and \(\lim _{r\searrow 0}h(r)=0\).

*h*is defined as follows. Let

*X*be a metric space and \(E\subseteq X\). For \(\delta >0\), we write

*h*-dimensional Hausdorff measure \({\mathcal {H}}^h(E)\) of

*E*is now defined by

*h*is defined as follows. For \(x\in X\) and \(r>0\), let

*C*(

*x*,

*r*) denote the closed ball in

*X*with centre at

*x*and radius equal to

*r*, and for \(\delta >0\), write

*h*-dimensional prepacking measure \(\overline{{\mathcal {P}}}^h(E)\) of

*E*is now defined by

*h*-dimensional packing measure \({\mathcal {P}}^{t}(E)\) of

*E*is defined as follows

### 2.3 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, \(\delta \) say, the Hewitt–Stromberg measures are defined using packings of balls with the *same* diameter \(\delta \).

*X*be a metric space and \(E\subseteq X\). 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

*h*. For a metric space

*X*and \(E\subseteq X\), we define the lower and upper

*h*-dimensional Hewitt–Stromberg pre-measures, denote by \(\overline{{\mathcal {U}}}^{h}\) and \(\overline{{\mathcal {V}}}^{h}\), respectively, by

*h*-dimensional Hewitt–Stromberg measures, denote by \({{\mathcal {U}}}^{h}\) and \({{\mathcal {V}}}^{h}\), respectively, by

### Proposition 2.1

*h*be a dimension function. Then we have

*X*and all \(E\subseteq X\).

### Proof

This follows immediately from the definitions; see also [2, pp. 32–36]. \(\square \)

### 2.4 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

*h*be a continuous dimension function.

- (1)A typical compact metric space \(X\in K_{\textsf {GH}}\) satisfies$$\begin{aligned} { {\mathcal {H}}}^{h}(X) = { {\mathcal {U}}}^{h}(X) = \overline{ {\mathcal {U}}}^{h}(X) = 0. \end{aligned}$$
- (2)A typical compact metric space \(X\in K_{\textsf {GH}}\) satisfiesfor all non-empty open subsets$$\begin{aligned} {\mathcal {V}}^{h}(U) = \overline{{\mathcal {V}}}^{h}(U) = {\mathcal {P}}^{h}(U) = \infty \end{aligned}$$
*U*of*X*. In particular, a typical compact metric space \(X\in K_{\textsf {GH}}\) satisfies$$\begin{aligned} {\mathcal {V}}^{h}(X) = \overline{{\mathcal {V}}}^{h}(X) = {\mathcal {P}}^{h}(X) = \infty . \end{aligned}$$

*h*and a subset

*E*of a metric space

*X*, we define the

*h*-dimensional box counting function \(f_{E}^{h}:(0,\infty )\rightarrow [0,\infty ]\) of

*E*by

*h*-dimensional box counting function \(f_{E}^{h}(t) = M_{e^{-t}}(E)\,h(2e^{-t})\) in (2.2) is obtained from the

*h*-dimensional box counting function \(F_{E}^{h}(r) = M_{r}(E)\,h(2r)\) in Sect. 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 \(t\rightarrow \infty \) when forming averages than letting \(r\searrow 0\)). Using the notation from (2.1), the Hewitt–Stromberg pre-measures of

*X*are now given by

*h*-dimensional box counting function \(f_{X}^{h}(t)\) of a typical compact metric space \(X\in K_{\textsf {GH}}\) diverges in the worst possible way as \(t\rightarrow \infty \). Below we analyse this divergence in detail using average procedures known as average systems.

### 2.5 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 \(\Pi =(\Pi _{t})_{t\ge t_{0}}\) with \(t_{0}>0\) such that:

- (i)
\(\Pi _{t}\) is a finite Borel measure on \([t_{0},\infty )\);

- (ii)
\(\Pi _{t}\) has compact support;

- (iii)
The Consistency Condition: If \(f:[t_{0},\infty )\rightarrow [0,\infty )\) is a positive measurable function and there is a real number

*a*such that \( f(t)\rightarrow a\,\,\,\,\text {as } t\rightarrow \infty , \) then \( \int f\,d\Pi _{t}\rightarrow a\,\,\,\,\text {as } t\rightarrow \infty . \)

*f*by

Applying averaging systems to the box counting function \(f_{E}^{h}(t)\) 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 \(\Pi =(\Pi _{t})_{t\ge t_{0}}\) be an averaging system. For a metric space

*X*and \(E\subseteq X\), we define the lower and upper \(\Pi \)-average

*h*-dimensional Hewitt–Stromberg pre-measures of

*E*by

*h*-dimensional Hewitt–Stromberg measures of

*E*by

*X*a metric space and we let \(\Pi \) denote the average system defined by \(\Pi =(\delta _{t})_{t\ge 1}\) (where \(\delta _{t}\) denotes the Dirac measure concentrated at

*t*), then clearly

*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

*h*be a dimension function and let \(\Pi \) be an averaging system. Then we have

*X*and all \(E\subseteq X\).

### Proof

The statement is not difficult to prove and we have therefore decided to omit the proof. \(\square \)

### 2.6 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.

### Definition

(*h**-closure-stable*) Let *h* be a dimension function. An averaging system \(\Pi \) is called *h*-closure-stable if \(\overline{{\mathcal {U}}}_{\Pi }^{h}(\overline{E})=\overline{{\mathcal {U}}}_{\Pi }^{h}(E)\) and \(\overline{{\mathcal {V}}}_{\Pi }^{h}(\overline{E})=\overline{{\mathcal {V}}}_{\Pi }^{h}(E)\) for all metric spaces *X* and all \(E\subseteq X\).

We immediately note that two important classes of averaging system are closure-stable. Namely, Proposition 2.3 (below) shows that the trivial averaging system \(\Pi =(\delta _{t})_{t\ge 1}\) is *h*-closure-stable for any dimension function, and Proposition 2.4 (below) shows that if \(\Pi =(\Pi _{t})_{t\ge t_{0}}\) is an averaging system and each measure \(\Pi _{t}\) has a continuous density with respect to Lebesgue measure, then \(\Pi \) is *h*-closure-stable for any continuous dimension function.

### Proposition 2.3

The averaging system \(\Pi =(\delta _{t})_{t\ge 1}\) (where \(\delta _{t}\) 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

*t*there is a measurable function \(\pi _{t}:[t_{0},\infty )\rightarrow [0,\infty )\) satisfying

*B*and such that if we write \(T_{t}=\sup {\text {supp}}\pi _{t}\), then

- (i)
\({\text {supp}}\pi _{t}=[t_{0},T_{t}]\);

- (ii)
\(\pi _{t}\) is continuous on \([t_{0},T_{t}]\);

- (iii)
\(\pi _{t}>0\) on \([t_{0},T_{t}]\).

*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 \( f_{X}^{h}(t) = M_{e^{-t}}(X)\,h(2e^{-t}) \) of a typical compact metric space *X* divergent as \(t\rightarrow \infty \), but it is so irregular that it remains spectacularly divergent as \(t\rightarrow \infty \) 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.

### Theorem 2.5

*h*be a continuous dimension function and let \(\Pi \) be an averaging system.

- (1)A typical compact metric space \(X\in K_{\textsf {GH}}\) satisfies$$\begin{aligned} \overline{ {\mathcal {U}}}_{\Pi }^{h}(X) = 0. \end{aligned}$$
- (2)A typical compact metric space \(X\in K_{\textsf {GH}}\) satisfiesfor all non-empty open subsets$$\begin{aligned} \overline{{\mathcal {V}}}_{\Pi }^{h}(U) = \infty \end{aligned}$$
*U*of*X*. - (3)If, in addition, \(\Pi \) is
*h*-closure-stable, then a typical compact metric space \(X\in K_{\textsf {GH}}\) satisfiesfor all non-empty open subsets$$\begin{aligned} {{\mathcal {V}}}_{\Pi }^{h}(U) = \infty \end{aligned}$$*U*of*X*.

The proof of Theorem 2.5 is given in Sects. 4–6. Section 4 contains various preliminary auxiliary results; the proof of Theorem 2.5.(1) is given in Sect. 5, and the proofs of Theorem 2.5.(2)–(3) are given in Sect. 6.

Note that if we apply Theorem 2.5 to the trivial average system \(\Pi \) defined by \(\Pi =(\delta _{t})_{t\ge 1}\), 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 \( f_{X}^{h}(t) = M_{e^{-t}}(X)\,h(2e^{-t}) \); this is the contents of Sect. 3 below.

## 3 Hölder and Cesaro averages of Hewitt–Strimborg measurers of a typical compact metric space

*X*. For \(a>0\) and a positive measurable function \(f:(a,\infty )\rightarrow [0,\infty )\), we define \(Mf:(a,\infty )\rightarrow [0,\infty )\) by

*n*, we now define the lower and upper

*n*’th order Hölder averages of

*f*by

*n*, we now define the lower and upper

*n*’th order Cesaro averages of

*f*by

*n*, define the averaging system \(\Pi _{n}^{\textsf {H}} = (\Pi _{n,t}^{\textsf {H}})_{t\ge a}\) by

*B*of \([a,\infty )\), then

*n*, define the averaging system \(\Pi _{n}^{\textsf {C}}=(\Pi _{n,t}^{\textsf {C}})_{t\ge a}\) by

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 \( f_{X}^{h}(t) = M_{e^{-t}}(X)\,h(2e^{-t})\). 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 \(n\ge 0\). We define the lower and upper

*n*’th order average Hölder Hewitt–Stromberg measures of a subset

*E*of

*X*, denoted \( {\mathcal {U}}_{n}^{\textsf {H},h}(E)\) and \( {\mathcal {V}}_{n}^{\textsf {H},h}(E)\), by

*n*’th order average Cesaro Hewitt–Stromberg measures of a subset

*E*of

*X*, denoted \( {\mathcal {U}}_{n}^{\textsf {C},h}(E)\) and \( {\mathcal {V}}_{n}^{\textsf {C},h}(E)\), by

*all*orders are sufficiently powerful to “smoothen out” the behaviour of the box counting function \( f_{X}^{h}(t) = M_{e^{-t}}(X)\,h(2e^{-t})\) as \(t\rightarrow \infty \).

### Theorem 3.1

*h*be a continuous dimension function. A typical compact metric space \(X\in K_{\textsf {GH}}\) satisfies

*U*of

*X*and all \(n\in \mathbb N\cup \{0\}\).

### Proof

This statement follows immediately from Theorem 2.5. \(\square \)

## 4 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 \(\Pi =(\Pi _{t})_{t\ge t_{0}}\) is an averaging system and each measure \(\Pi _{t}\) has a continuous density with respect to Lebesgue measure, then \(\Pi \) is *h*-closure-stable for any continuous dimension function *h*.

*X*is a metric space and \(E\subseteq X\), then the packing number \(M_{r}(E)\) of

*E*is defined by

*E*by

### Lemma 4.1

- (1)
The function \(N_{r}:K_{\textsf {GH}} \rightarrow \mathbb R\) is lower semi-continuous for all \(r>0\).

- (2)
The function \(M_{r}:K_{\textsf {GH}} \rightarrow \mathbb R\) is upper semi-continuous for all \(r>0\).

- (3)
We have \(M_{r}(X)\le N_{r}(X)\le M_{\frac{r}{3}}(X)\) for all \(r>0\) and all \(X\in K_{\textsf {GH}}\).

Secondly, we prove Lemma 4.2 providing a useful technique for establishing lower bounds for the average *h*-dimensional Hewitt–Stromberg measures associated with *h*-closure-stable average systems. Recall (see Sect. 2.6), that if *h* is a dimension function, then an average system \(\Pi \) is called *h*-closure-stable if \( \overline{{\mathcal {U}}}_{\Pi }^{h}(\overline{E}) = \overline{{\mathcal {U}}}_{\Pi }^{h}(E)\) and \( \overline{{\mathcal {V}}}_{\Pi }^{h}(\overline{E}) = \overline{{\mathcal {V}}}_{\Pi }^{h}(E)\) for all metric spaces *X* and all \(E\subseteq X\). We can now state Lemma 4.2.

### Lemma 4.2

*h*be a continuous dimension function and let \(\Pi =(\Pi _{t})_{t\ge t_{0}}\) be an

*h*-closure-stable averaging system. Let

*X*be a complete metric space and let

*C*be a compact subset of

*X*. Fix \(c\ge 0\). Then the following statements hold.

- (1)
If \( \overline{{\mathcal {U}}}_{\Pi }^{h}(V\cap C) \ge c \) for all open subsets

*V*of*X*with \(V\cap C\not =\varnothing \), then \( {{\mathcal {U}}}_{\Pi }^{h}(C) \ge c\). - (2)
If \( \overline{{\mathcal {V}}}_{\Pi }^{h}(V\cap C) \ge c \) for all open subsets

*V*of*X*with \(V\cap C\not =\varnothing \), then \( {{\mathcal {V}}}_{\Pi }^{h}(C) \ge c\).

### Proof

*V*of

*X*with \(V\cap C\not =\varnothing \). We must now show that \( {{\mathcal {U}}}_{\Pi }^{h}(C) \ge c\). Let \((E_{i})_{i}\) be a countable family of subsets of

*X*with \(C\subseteq \cup _{i}E_{i}\). Since \(C = \cup _{i}E_{i} \subseteq \cup _{i}\overline{{E_{i}}}\), it follows from Baire’s category theorem that there is an index \(i_{0}\) and an open subset

*W*of

*X*such that \(C\cap W\not =\varnothing \) and \(C\cap W\subseteq \overline{{E_{i_{0}}}}\). We therefore conclude that \( \overline{{\mathcal {U}}}_{\Pi }^{h}(\,\overline{E_{i_{0}}}\,) \ge \overline{{\mathcal {U}}}_{\Pi }^{h}(C\cap W) \ge c\). Since \(\Pi \) is

*h*-closure-stable, it follows from this that

*X*with \(C\subseteq \cup _{i}E_{i}\), shows that \({\mathcal {U}}_{\Pi }^{h}(E) = \inf _{E\subseteq \cup _{i=1}^\infty E_i}\sum _{i=1}^\infty \overline{{\mathcal {U}}}_{\Pi }^h(E_i)\ \ge c\).

(2) The proof of this statement is identical to the proof of the statement in Part (1) and is therefore omitted. \(\square \)

Thirdly, and finally, we prove Proposition 2.4 from Sect. 2. We start with a small lemma.

### Lemma 4.3

*X*and all \(E\subseteq X\).

### Proof

Write \(\delta =1-u\in (0,1)\). It follows from the definition of the packing number \(M_{r}(\,\overline{E}\,)\) that we can find a family \((C(x_{i},r))_{i=1}^{M_{r}(\,\overline{E}\,)}\) of pairwise disjoint closed balls \(C(x_{i},r)\) in *X* with radii equal to *r* and centres \(x_{i}\in \overline{E}\). Since \(x_{i}\in \overline{E}\), there is a point \(y_{i}\in E\) such that \(y_{i}\in B(x_{i},\delta r)\), whence \(C(y_{i},(1-\delta )r) \subseteq C(x_{i},r)\). As the family \((C(x_{i},r))_{i=1}^{M_{r}(\,\overline{E}\,)}\) consists of pairwise disjoint balls, we therefore conclude that \((C(y_{i},(1-\delta )r))_{i=1}^{M_{r}(\,\overline{E}\,)}\) is a family of pairwise disjoint closed balls in *X* with radii equal to \((1-\delta )r\) and centres \(y_{i}\in E\). This clearly implies that \( M_{r}(\,\overline{E}\,) \le M_{(1-\delta )r}(E) = M_{ur}(E)\). \(\square \)

We can now prove Proposition 2.4.

### Proof of Proposition 2.4

Let *X* be a metric space and \(E\subseteq X\). It is clear that \( \overline{{\mathcal {U}}}_{\Pi }^{h}(E) \le \overline{{\mathcal {U}}}_{\Pi }^{h}(\overline{E})\) and \( \overline{{\mathcal {V}}}_{\Pi }^{h}(E) \le \overline{{\mathcal {V}}}_{\Pi }^{h}(\overline{E})\), and it therefore suffices to prove that \( \overline{{\mathcal {U}}}_{\Pi }^{h}(\overline{E}) \le \overline{{\mathcal {U}}}_{\Pi }^{h}(E)\) and \( \overline{{\mathcal {V}}}_{\Pi }^{h}(\overline{E}) \le \overline{{\mathcal {V}}}_{\Pi }^{h}(E)\). We will now prove these inequalities. Let \(\varepsilon >0\). Fix \(t\ge t_{0}\). Next we define three numbers \(\rho _{t}\), \(\delta _{t}\) and \(u_{t}\) as follows.

*Definition of*\(\rho _{t}\). Write \( K_{t} = \{ (u,v)\,|\, u\ge t_{0}, \,\, v\ge 0, \,\, t_{0}\le u-v\le T_{t} \}\) and define \(D_{t}:K_{t}\rightarrow [0,\infty )\) by \(D_{t}(u,v)=\frac{\pi _{t}(u-v)}{\pi _{t}(u)}\). 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 number \(\rho _{t}>0\) such that if \((u',v'),(u'',v'')\in K_{t}\) and \(|(u',v')-(u'',v'')|\le \rho _{t}\), then \( |D_{t}(u',v')-D_{t}(u'',v'')|\le \varepsilon \). In particular, this implies that if \((u,v)\in K_{t}\) and \(v\le \rho _{t}\), then \(|(u,v)-(u,0)|=v\le \rho _{t}\), and so \( |D_{t}(u,v)-D_{t}(u,0)|\le \varepsilon \), whence

*Definition of*\(\delta _{t}\). Since

*h*is continuous, and therefore uniformly continuous on compact subintervals of \((0,\infty )\), we can find a positive real number \(\delta _{t}>0\), such that if \(s',s''\in [t_{0},T_{t}]\) with \(|s'-s''|\le \delta _{t}\), then

*Definition of*\(u_{t}\). We can clearly choose a positive number \(0<u_{t}<1\) such that

*s*. Also, if \(s\in [t_{0},T_{t}]\), then we conclude from this and Lemma 4.3 that

## 5 Proof of Theorem 2.5.(1)

*h*is a dimension function and \(X\in K_{\textsf {GH}}\), then we define the function \(f_{X}^{h}:(0,\infty )\rightarrow (0,\infty )\) by

*h*, an averaging system \(\Pi =(\Pi _{t})_{t\ge t_{0}}\), and \(t,c>0\), write

### Lemma 5.1

Let *h* be a dimension function and let \(\Pi =(\Pi _{t})_{t\ge t_{0}}\) be an averaging system. Fix \(t,c>0\). Then the set \(L_{t,c}^{h,\Pi }\) is open in \( K_{\textsf {GH}}\).

### Proof

*F*is closed in \(K_{\textsf {GH}}\). In order to show this, we fix a sequence \((X_{n})_{n}\) in

*F*and \(X\in K_{\textsf {GH}}\) with \(X_{n}\rightarrow X\). We must now prove that \(X\in F\), i.e. we must prove that \( \int f_{X}^{h} \,d\Pi _{t} \ge c\). We prove this inequality below. For brevity define functions \(\varphi ,\varphi _{n}:[t_{0},\infty )\rightarrow [0,\infty )\) by

We now prove the following three claims.

### Claim 1

We have \(\int \sup _{n}\varphi _{n}\,d\Pi _{t}<\infty \).

### Proof of Claim 1

The measure \(\Pi _{t}\) has compact support, and we can therefore find \(T_{0}\ge t_{0}\), such that \({\text {supp}}\Pi _{t}\subseteq [t_{0},T_{0}]\). It follows from Lemma 4.1 that \(M_{e^{-T_{0}}}\) is upper semi-continuous, and so \(\limsup _{n}M_{e^{-T_{0}}}(X_{n})\le M_{e^{-T_{0}}}(X)\). In particular, this implies that there is a constant *K* such that \(M_{e^{-T_{0}}}(X_{n})\le K\) for all *n*. For positive integers *n* and \(s\in [t_{0},T_{0}]\) we therefore conclude that \( \varphi _{n}(s) = M_{e^{-s}}(X_{n})\,h(2e^{-s}) \le M_{e^{-T_{0}}}(X_{n})\,h(2e^{-t_{0}}) \le K\,h(2e^{-t_{0}}) \). Finally, since \({\text {supp}}\Pi _{t}\subseteq [t_{0},T_{0}]\), it therefore follows that \(\int \sup _{n}\varphi _{n}\,d\Pi _{t} = \int _{t_{0}}^{T_{0}} \sup _{n}\varphi _{n}\,d\Pi _{t} \le K\,h(2e^{-t_{0}}) \, \Pi _{t}([t_{0},T_{0}]) <\infty \) This completes the proof of Claim 1.

### Claim 2

We have \(c\le \int \limsup _{n}\varphi _{n}\,d\Pi _{t}\).

### Proof of Claim 2

*n*, whence

### Claim 3

For all \(s\ge t_{0}\), we have \(\limsup _{n}\varphi _{n}(s)\le \varphi (s)\), and so \(\int \limsup _{n}\varphi _{n}\,d\Pi _{t}\le \int \varphi \,d\Pi _{t}\).

### Proof of Claim 3

This follows from the fact that \(M_{r}:K_{\textsf {GH}}\rightarrow \mathbb R\) 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 \( c \le \int \limsup _{n}\varphi _{n}\,d\Pi _{t} \le \int \varphi \,d\Pi _{t} = \int f_{X}^{h} \,d\Pi _{t}\). \(\square \)

### Proposition 5.2

*h*be a dimension function and let \(\Pi =(\Pi _{t})_{t\ge t_{0}}\) be an averaging system.

- (1)For \(c\in \mathbb R^{+}\), writeThen \(T_{c}\) is co-meagre.$$\begin{aligned} T_{c} = \left\{ X\in K_{\textsf {GH}} \,\Big |\, \overline{{\mathcal {U}}}_{\Pi }^{h}(X)\le c \right\} . \end{aligned}$$
- (2)WriteThen$$\begin{aligned} T = \left\{ X\in K_{\textsf {GH}} \,\Big |\, \overline{{\mathcal {U}}}_{\Pi }^{h}(X)=0 \right\} . \end{aligned}$$
*T*is co-meagre.

### Proof

### Claim 1

\(G_{u}\) is open in \(K_{\textsf {GH}}\).

### Claim 2

\(G_{u}\) is dense in \(K_{\textsf {GH}}\).

### Proof of Claim 2

We first prove that \(\{ X\in K_{\textsf {GH}} \,|\, X \text { is finite} \} \subseteq G_{u}\). Indeed, if *X* is a finite metric space, then it is clear that \(f_{X}^{h}(t) = M_{e^{-t}}(X)\,h(2e^{-t}) \rightarrow 0\), and the consistency condition therefore implies that \(\int f_{X}^{h}\,d\Pi _{t}\rightarrow 0\). It follows from this that there is a number *t* with \(t>u\) and \(\int f_{X}^{h}\,d\Pi _{t}<c\), whence \(X\in L_{t,c}^{h,\Pi }\subseteq G_{u}\). This proves that \(\{ X\in K_{\textsf {GH}} \,|\, X \text { is finite} \} \subseteq G_{u}\).

Next, since \(\{ X\in K_{\textsf {GH}} \,|\, X \text { is finite} \} \subseteq G_{u}\) and \( \{ X\in K_{\textsf {GH}} \,|\, X \text { is finite} \}\) is dense in \(K_{\textsf {GH}}\), we conclude that \(G_{u}\) is dense in \(K_{\textsf {GH}}\). This completes the proof of Claim 2.

### Claim 3

\(\cap _{u\in {\mathbb {Q}}^{+}}G_{u}\subseteq T_{c}\).

### Proof of Claim 3

Let \(X\in \cap _{u\in {\mathbb {Q}}^{+}}G_{u}\). We must now show that \(\overline{{\mathcal {U}}}_{\Pi }^{h}(X)\le c\). Since \(X \in \cap _{u\in {\mathbb {Q}}^{+}}G_{u} \subseteq \cap _{n}G_{n} \), we conclude that for each positive integer *n*, we can find a positive number \(t_{n}\) with \(t_{n}>n\) such that \(X\in L_{t_{n},c}^{h}\), whence \(\int f_{X}^{h}\,d\Pi _{t_{n}}<c\). It follows immediately from this that \(\overline{{\mathcal {U}}}_{\Pi }^{h}(X) = \liminf _{t}\int f_{X}^{h}\,d\Pi _{t} \le \liminf _{n}\int f_{X}^{h}\,d\Pi _{t_{n}} \le c\), and so \(X\in T_{c}\). This completes the proof of Claim 3.

(2) This statement follows immediately from Part (1) since \(T=\cap _{c\in {\mathbb {Q}}^{+}}T_{c}\). \(\square \)

We can now prove Theorem 2.5.(1).

## 6 Proof of Theorem 2.5.(2)–(3)

*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 \(g_{X}^{h}:(0,\infty )\rightarrow (0,\infty )\) by

*h*, an averaging system \(\Pi =(\Pi _{t})_{t\ge t_{0}}\) and \(r,t,c>0\), write

### Lemma 6.1

Let *h* be a dimension function and let \(\Pi =(\Pi _{t})_{t\ge t_{0}}\) be an averaging system. Fix \(t,c>0\). Then the set \(\Lambda _{t,c}^{h,\Pi }\) is open in \( K_{\textsf {GH}}\).

### Proof

*F*is closed in \(K_{\textsf {GH}}\). In order to show this, we fix a sequence \((X_{n})_{n}\) in

*F*and \(X\in K_{\textsf {GH}}\) with \(X_{n}\rightarrow X\). We must now prove that \(X\in F\), i.e. we must prove that \( \int g_{X}^{h} \,d\Pi _{t} \le c\). We prove this inequality below. For brevity define functions \(\varphi ,\varphi _{n}:[t_{0},\infty )\rightarrow [0,\infty )\) by

We now prove the following two claims.

### Claim 1

We have \(\int \liminf _{n}\varphi _{n}\,d\Pi _{t}\le c\).

### Proof of Claim 1

Since \(X_{n}\in F\), we conclude that \( \int \varphi _{n} \,d\Pi _{t} = \int g_{X_{n}}^{h}\,d\Pi _{t} \le c \) for all *n*, whence \( \liminf _{n} \int \varphi _{n} \,d\Pi _{t} \le c \). It follows immediately from this and Fatou’s Lemma that \(\int \liminf _{n}\varphi _{n} \,d\Pi _{t} \le \liminf _{n}\int \varphi _{n} \,d\Pi _{t} \le c\). This completes the proof of Claim 1.

### Claim 2

For all \(s\ge t_{0}\), we have \(\varphi (s)\le \liminf _{n}\varphi _{n}(s)\), and so \(\int \varphi \,d\Pi _{t}\le \int \liminf _{n}\varphi _{n}\,d\Pi _{t}\).

### Proof of Claim 3

This follows from the fact that \(N_{r}:K_{\textsf {GH}}\rightarrow \mathbb R\) 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 \( \int g_{X}^{h} \,d\Pi _{t} = \int \varphi \,d\Pi _{t} \le \int \liminf _{n}\varphi _{n}\,d\Pi _{t} \le c\). \(\square \)

### Proposition 6.2

Let *h* be a dimension function and let \(\Pi =(\Pi _{t})_{t\ge t_{0}}\) be an averaging system. Fix \(r,t,c>0\). Then the set \(L_{r,t,c}^{h,\Pi }\) is open in \(K_{\textsf {GH}}\).

### Proof

Let \(X\in L_{r,t,c}^{h,\Pi }\) and let \(d_{X}\) denote the metric in *X*. We must now find \(\rho >0\) such that \(B(X,\rho )\subseteq L_{r,t,c}^{h}\).

*N*and

*X*is compact, we conclude that there is a point \(x_{0}\in X\) such that \(\Phi (x_{0})=\sup _{x\in X}\Phi (x)\). For brevity write \(r_{0}=\Phi (x_{0})=\sup _{x\in X}\Phi (x)\), and note that since \(x_{0}\in X=\cup _{i}B(x_{i},r)\), we can find \(i_{0}\) with \(x_{0}\in B(x_{i_{0}},r)\), whence

*Y*. Since \(d_{\textsf {GH}}(X,Y)<\rho \), it follows that we may assume that there is a complete metric space \((Z,d_{Z})\) with \(X,Y\subseteq Z\) and \(d_{\textsf {H}}(X,Y)<\rho \) such that \(d_{X}(x',x'') = d_{Z}(x',x'')\) for all \(x',x''\in X\), and \(d_{Y}(y',y'') = d_{Z}(y',y'')\) for all \(y',y''\in Y\). 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 \(\delta \) and centre at \(y\in Y\) by \(B_{Y}(y,\delta )\), i.e. \(B_{Y}(y,\delta ) = \{y'\in Y\,|\,d_{Y}(y,y')<\delta \}\), and we will denote the open ball in

*Z*with radius equal to \(\delta \) and centre at \(z\in Z\) by \(B_{Z}(z,\delta )\), i.e. \(B_{Z}(z,\delta ) = \{z'\in X\,|\,d_{X}(z,z')<\delta \}\). We must now show that \(Y\in L_{r,t,c}^{h,\Pi }\). Since \(d_{\textsf {H}}(X,Y)<\rho \), we conclude that for each

*i*, there is a point \(y_{i}\in Y\) with \(d_{Z}(x_{i},y_{i})<\rho \). Next, put

### Claim 1

\(Y=\cup _{i}B_{Y}(y_{i},r)\).

### Proof of Claim 1

It is clear that \(\cup _{i}B_{Y}(y_{i},r)\subseteq Y\). In order to prove the reverse inclusion, we let \(y\in Y\). Since \(d_{\textsf {H}}(X,Y)<\rho \), we conclude that there is a point \(x\in X\) with \(d_{Z}(x,y)<\rho \). Also, since \(\min _{i}d_{X}(x,x_{i})=\Phi (x)\le r_{0}\), we deduce that there is an index *j* with \(d_{X}(x,x_{j})\le r_{0}\). Finally, it follows from the definition of \(y_{j}\) that \(d_{Z}(x_{j},y_{j})<\rho \). Hence \(d_{Y}(y,y_{j}) = d_{Z}(y,y_{j}) \le d_{Z}(y,x) + d_{Z}(x,x_{j}) + d_{Z}(x_{j},y_{j}) = d_{Z}(y,x) + d_{X}(x,x_{j}) + d_{Z}(x_{j},y_{j}) < \rho +r_{0}+\rho = 2\rho +r_{0} \le r\), and so \(y\in B_{Y}(y_{j},r)\subseteq \cup _{i}B_{Y}(y_{i},r)\). This completes the proof of Claim 1.

### Claim 2

\(K_{i}\subseteq B(y_{i},r)\) for all *i*.

### Proof of Claim 2

### Claim 3

\(K_{i}\in \Lambda _{t_{i},c}^{h,\Pi }\) for all *i*.

### Proof of Claim 3

It is clear that \(K_{i}\) is a closed subset of *Y* and so \(K_{i}\in K_{\textsf {GH}}\).

Combining (6.14) and (6.15), we immediately conclude that \(d_{\textsf {H}}(C_{i},K_{i}) = \max ( \sup _{x\in C_{i}}{\text {dist}}(x,K_{i}), \sup _{y\in K_{i}}{\text {dist}}(y,C_{i}) ) \le \rho \le \tfrac{\rho _{i}}{2} < \rho _{i}\), whence \(K_{i}\in B(C_{i},\rho _{i}) \subseteq \Lambda _{t_{i},c}^{h,\Pi }\). This completes the proof of Claim 3.

It follows immediately from Claims 1–3 that \(Y\in L_{r,t,c}^{h}\). \(\square \)

### Proposition 6.3

*h*be a continuous dimension function and let \(\Pi =(\Pi _{t})_{t\ge t_{0}}\) be an averaging system.

- (1)For \(c\in \mathbb R^{+}\), writeThen \(T_{c}\) is co-meagre.$$\begin{aligned} T_{c} = \left\{ X\in K_{\textsf {GH}} \,\Big |\, \overline{{\mathcal {V}}}_{\Pi }^{h}(U)\ge c\quad \text {for all open subsets } U \text { of } X \text { with } U\not =\varnothing \right\} . \end{aligned}$$
- (2)WriteThen$$\begin{aligned} T = \left\{ X\in K_{\textsf {GH}} \,\Big |\, \overline{{\mathcal {V}}}_{\Pi }^{h}(U)=\infty \quad \text {for all open subsets } U \text { of } X \text { with } U\not =\varnothing \right\} . \end{aligned}$$
*T*is co-meagre. - (3)WriteIf, in addition, \( \overline{{\mathcal {V}}}_{\Pi }^{h}(\overline{E}) = \overline{{\mathcal {V}}}_{\Pi }^{h}(E)\) for all metric spaces and all \(E\subseteq X\), then$$\begin{aligned} S = \left\{ X\in K_{\textsf {GH}} \,\Big |\, {\mathcal {V}}_{\Pi }^{h}(U)=\infty \quad \text {for all open subsets } U \text { of } X \text { with } U\not =\varnothing \right\} . \end{aligned}$$
*S*is co-meagre.

### Proof

### Claim 1

\(G_{r,u}\) is open in \(K_{\textsf {GH}}\).

### Claim 2

\(G_{r,u}\) is dense in \(K_{\textsf {GH}}\).

### Proof of Claim 2

Let \(X\in K_{\textsf {GH}}\) and \(\rho >0\). Also, let \(d_{X}\) denote the metric in *X*. We must now find \(Y\in G_{r,u}\) such that \(d_{\textsf {GH}}(X,Y)<\rho \).

*X*is compact, we can find a finite subset

*E*of

*X*such that

*T*,

*t*such that \(T>t>u\), and for positive integers

*n*, define \(\varphi _{n}:(t_{0},\infty )\rightarrow (0,\infty )\) by

*N*with

*Y*by

*Y*with the supremum metric \(d_{Y}\) induced by \(d_{X}\) and \(|\cdot |\), i.e. \(d_{Y}(\,(x',z'), \,(x'',z'')\,) = \max ( \, d_{X}(x',x'') , \, |z'-z''| \, )\) for \(x',x''\in E\) and \(z',z''\in C\). It is clear that

*Y*is compact, and so \(Y\in K_{\textsf {GH}}\). Below we show that \(d_{\textsf {GH}}(X,Y)<\rho \) and \(Y\in G_{r,u}\).

*Proof of*\(d_{\textsf {GH}}(X,Y)<\rho \). Define \(f:E\rightarrow Y\) and \(g:Y\rightarrow Y\) by \(f(x)=(x,0)\) and \(g:Y\rightarrow Y\) by \(g(x,z)=(x,z)\). It is clear that *f* and *g* are isometries and we therefore conclude that \( d_{\textsf {GH}}(E,Y) \le d_{\textsf {H}}(f(E),g(Y)) = d_{\textsf {H}}(E\times \{0\}\},E\times C) \le \sup _{z\in C}|z| = \delta _{0} \le \tfrac{\rho }{2}\), whence \( d_{\textsf {GH}}(X,Y) \le d_{\textsf {GH}}(X,E) + d_{\textsf {GH}}(E,Y)< d_{\textsf {H}}(X,E) + \tfrac{\rho }{2} < \tfrac{\rho }{2} + \tfrac{\rho }{2} =\rho \). This completes the proof of \(d_{\textsf {GH}}(X,Y)<\rho \).

*Proof of*\(Y\in G_{r,u}\). Write \(E=\{x_{1},\ldots ,x_{M}\}\) and put

Indeed, it is clear that \(Y = E\times C = \cup _{i}(\{x_{i}\}\times C) = \cup _{i}B(y_{i},r)\); this proves (6.18).

It is also clear that \(C_{i} \subseteq B(y_{i},r)\) for all *i*; this proves (6.19).

It follows immediately from (6.18)–(6.20) that \(Y\in L_{r,t,c}^{h,\Pi }\subseteq G_{r,u}\). This completes the proof of Claim 2.

### Claim 3

\(\cap _{r,u\in {\mathbb {Q}}^{+}}G_{r,u}\subseteq T_{c}\).

### Proof of Claim 3

*U*is an open subset of

*X*with \(U\not =\varnothing \), then \(\overline{{\mathcal {V}}}_{\Pi }^{h}(U)\ge c\). We therefore let

*U*be an open subset of

*X*with \(U\not =\varnothing \), and proceed to show that \(\overline{{\mathcal {V}}}_{\Pi }^{h}(U)\ge c\). Since

*U*is non-empty and open there is \(x_{0}\in U\) and \(r_{0}>0\) with \(B(x_{0},r_{0})\subseteq U\). Next, since \(X \in \cap _{r,u\in {\mathbb {Q}}^{+}}G_{r,u} \subseteq \cap _{n}G_{\frac{r_{0}}{2},n} \), we conclude that for each positive integer

*n*, we can find a positive real number \(t_{n}\) with \(t_{n}>n\) such that \(X\in L_{\frac{r_{0}}{2},t_{n},c}^{h,\Pi }\). In particular, this implies that there is a positive integer \(N_{n}\) and

*s*, we conclude from this combined with the fact that \(C_{n,i_{n}}\in \Lambda _{t_{n,i_{n}},c}^{h,\Pi }\), that \(\int f_{U}^{h}\,d\Pi _{t_{n,i_{n}}} \ge \int g_{U}^{h}\,d\Pi _{t_{n,i_{n}}} \ge \int g_{C_{n,i_{n}}}^{h}\,d\Pi _{t_{n,i_{n}}} \ge c\). Finally, since \(t_{n,i_{n}}> t_{n}>n\) and so \(t_{n,i_{n}}\rightarrow \infty \), we deduce from the previous inequality that \(\overline{{\mathcal {V}}}_{\Pi }^{h}(U) = \limsup _{t}\int f_{U}^{h}\,d\Pi _{t} \ge \limsup _{n}\int f_{U}^{h}\,d\Pi _{t_{n,i_{n}}} \ge c\). This completes the proof of Claim 3.

(2) This statement follows immediately from Part (1) since clearly \(T=\cap _{c\in {\mathbb {Q}}^{+}}T_{c}\).

*U*is an open subset of

*X*with \(U\not =\varnothing \), then \({\mathcal {V}}_{\Pi }^{h}(U)=\infty \). We therefore let

*U*be an open subset of

*X*with \(U\not =\varnothing \), and proceed to show that \({\mathcal {V}}_{\Pi }^{h}(U)=\infty \). Since

*U*is non-empty and open there is \(x\in U\) and \(r>0\) such that \(B_{X}(x,r)\subseteq U\). In particular, if we write \(C=\overline{B(x,\tfrac{r}{2})}\), then

*C*is compact and \(C\subseteq B(x,r)\subseteq U\). Next, we prove the following claim.

### Claim 4

If *V* is an open subset of *X* with \(V\cap C\not =\varnothing \), then \(\overline{{\mathcal {V}}}_{\Pi }^{h}(V\cap C)=\infty \).

### Proof of Claim 4

*V*be an open subset of

*X*with \(V\cap C\not =\varnothing \). We must now show that \(\overline{{\mathcal {V}}}_{\Pi }^{h}(V\cap C)=\infty \). As \(V\cap C\not =\varnothing \), it is possible to choose \(y\in V\cap C\). Since \(y\in V\) and

*V*is open, we can choose \(\varepsilon >0\) such that \(B(y,\varepsilon )\subseteq V\). Next, since \(y\in C = \overline{B(x,\tfrac{r}{2})}\), we choose \(z\in B(x,\tfrac{r}{2})\) such that \(z\in B(y,\varepsilon )\). Finally, since \(z\in B(x,\tfrac{r}{2})\cap B(y,\varepsilon )\), we can find \(\delta >0\) with \(B(z,\delta )\subseteq B(x,\tfrac{r}{2})\cap B(y,\varepsilon )\), whence \(B(z,\delta )\subseteq B(x,\tfrac{r}{2})\cap B(y,\varepsilon ) \subseteq C\cap V\), and so

Finally, it follows immediately from Claim 4 and Lemma 4.2 that \({\mathcal {V}}_{\Pi }^{h}(C)=\infty \), and since \(C\subseteq U\), this implies that \({\mathcal {V}}_{\Pi }^{h}(U)=\infty \). \(\square \)

We can now prove Theorem 2.5.(2)–(3).

## Notes

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