On the Hausdorff and packing measures of typical compact metric spaces

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


Introduction
Recall that a subset E of a metric space M is called co-meagre if its complement is meagre; also recall that if P is a property that the elements of M may have, then we say that a typical element x in M has property P if the set E = {x ∈ M | x has property P} is co-meagre, see Oxtoby [9] for more details. The purpose of this paper is to investigate the Hausdorff and packing measures of a typical compact metric space belonging to the Gromov-Hausdorff space K GH of all compact metric spaces; the precise definition of the Gromov-Hausdorff space K GH will be given below. The four most commonly used fractal dimensions of a metric space X are: the lower and upper box dimensions, denoted by dim B (X) and dim B (X), respectively; the Hausdorff dimension, denoted by dim H (X); and the packing dimension, denoted by dim P (X); the precise definitions will be given in Sect. 2.2. It is well-known that if X is a metric space, then these dimensions satisfy the following inequalities, We now return to the main question in this paper: what are the dimensions of a typical compact metric space? Rouyer [13] has very recently provided the following answer to this question. Theorem A. [13] A typical compact metric space X ∈ K GH satisfies dim H (X) = dim B (X) = 0 , Theorem A shows that the lower box dimension of a typical compact metric space is as small as possible and that the upper box dimension of a typical compact metric space is as big as possible. Other studies of typical compact sets show the same dichotomy. For example, Gruber [3] and Myjak & Rudnicki [8] proved 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.
For example, as an application of our main results we show that not only is the upper box dimension of a typical compact metric space X ∈ K GH equal to infinity (see Theorem A above), but even the smaller packing dimension is equal to infinity; this is the content of Theorem 1.1 below. Theorem 1.1. A typical compact metric space X ∈ K GH satisfies dim P (X) = ∞ .
While Theorems A and 1.1 study and compute the dimensions of typical compact metric spaces, we prove more general results investigating and computing not only the dimensions of typical compact metric spaces but also the exact values of the Hausdorff and packing measures of typical compact metric spaces, see Theorem 2.4. In fact, we prove even stronger results providing information about the so-  for X, Y ∈ K GH . It is well-known that (K GH , d GH ) is a complete metric space [11]. The reader is referred to [11,Chapter 10], for a detailed discussion of the Gromov-Hausdorff space and the Gromov-Hausdorff metric.

Hausdorff measure, packing measure and box dimensions
While the definitions of the Hausdorff and packing measures (and the Hausdorff and packing dimensions) and box 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; and thirdly, in order to provide a motivation for the Hewitt-Stromberg measures. Let X be a metric space and let d be the metric in X. For x ∈ X and r > 0, let C(x, r) denote the closed ball with centre at x and radius equal to r, i.e. C(x, r) = {y ∈ X | d(x, y) ≤ r}. The lower and upper box dimensions of a subset E of X are defined as follows. For r > 0, the covering number N r (E) and the packing number M r (E) of E are defined by (2.1) The lower and upper box dimensions, denoted by dim B (E) and dim B (E), respectively, are now defined by Next, we recall the definitions of the Hausdorff and packing measures. We start by recalling the definition of a dimension function.

Definition. (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 If t > 0 and h t denotes the dimension function defined by h t (r) = r t , then we will follow the traditional convention and write The reader is referred to Rogers' classical text [12] for an excellent and systematic discussion of the Hausdorff measures H h .
The packing measure with a dimension function h is defined as follows. For E ⊆ X and δ > 0, write The h-dimensional prepacking measure P h (E) of E is now defined by Finally, we define the h-dimensional packing measure P t (E) of E, as follows As above, we note that if t > 0 and h t denotes the dimension function defined by h t (r) = r t , then we will follow the traditional convention and write P ht (E) = P t (E) . Finally, the packing dimension dim P (E) is defined by The reader is referred to [2] for an excellent discussion of the Hausdorff dimension, the packing dimension and the box dimensions.

Hewitt-Stromberg measures
Hewitt-Stromberg measures were introduced by Hewitt & Stromberg in their classical textbook [6, (10.51)], and have subsequently been investigated further by, for example, [4,5,14], highlighting their fundamental importance in the study of local properties of fractals and products of fractals. In particular, Edgar's textbook [1, pp. 32-36], provides an informative and systematic introduction to the Hewitt-Stromberg measures and their importance in the study of local properties of fractals. The measures also appear explicitly in, for example, Pesin's monograph [10, 5.3], who discusses their important role in the study of dynamical systems and implicitly in Mattila's text [7]. 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. For a dimension function h, the Hewitt-Stromberg measures are defined as follows. For a metric space X and E ⊆ X, write We now define the lower and upper h-dimensional Hewitt-Stromberg measures, denoted 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.
for all metric spaces X and all E ⊆ X.

Hewitt-Stromberg measures of typical compact spaces
Our first main result computes the Hewitt-Stromberg measures of a typical compact metric space; this is the content of Theorem 2.2 below.

Theorem 2.2. (Hewitt-Stromberg measures of typical compact spaces) Let h be a continuous dimension function.
( The proof of Theorem 2.2 is given in Sect. 3 and Sects. 5-6; Section 3 contains a number of preliminary auxiliary results, and the proofs of the statements in Theorems 2.2.(1) and 2.2.
(2) are given in Sects. 5 and 6, respectively. For brevity write and note that While it follows from Theorem 2.2 that the set M positive is meagre, the set M positive is, nevertheless, dense in K GH . In fact, even the smaller sets N infinity and M infinity are dense in K GH ; this is the content of Theorem 2.3 below.

Theorem 2.3. Let h be a continuous dimension function. Then the set
The proof of Theorem 2.3 is given in Section 4. We now present several applications of Theorem 2.2. In Section 2.5 we apply Theorem 2.2 to find the Hausdorff and packing measures for a typical compact metric space, and in Section 2.6 we apply the results from Section 2.5 to find the packing dimension (and other dimensions) of a typical compact metric space.

Hausdorff and packing measures of typical compact spaces
Because of the importance of the Hausdorff measures and the packing measures, the following corollary of Theorem 2.2 seems worthwhile stating separately.

Theorem 2.4. (Hausdorff measures and packing measures of typical compact metric spaces) Let h be a continuous dimension function.
( for all non-empty open subsets U of X. In particular, a typical compact metric space X ∈ K GH satisfies Proof. This result follows immediately from Proposition 2.1 and Theorem 2.2.

Packing dimensions of typical compact spaces
As a further specialization of Theorem 2.4 we obtain the next result about the Hausdorff and packing dimensions of typical compact spaces. While the result in Theorem 2.5.(1) (saying that dim H (X) = 0 for a typical compact metric space X) has already been obtained by Rouyer [13] (see Theorem A in Section 1), we believe that it is instructive to present a simple proof based on Theorem 2.4.
Since it follows from Theorem 2.2 that the set {X ∈ K GH | P t (U ) = ∞ for all non-empty open subsets U of X } is co-meagre for all t > 0, we conclude from We also obtain the following corollary providing information about the lower box dimension of a typical compact space.
There are continuous dimension functions h satisfying (2.5) such that Remark. For brevity write The statement in Part (3) of Corollary 2.6 has recently been obtained by Rouyer [13]. However, since Part (1) in Corollary 2.6 shows that S is a subset of T , we deduce that the statement in Part (2) is stronger than Rouyer's result in Part (3). In fact, since Part (1) in Corollary 2.6 also shows that S, in general, is a proper subset of T , we conclude that the statement in Part (2), in general, is strictly stronger than Rouyer's result in Part (3).
Proof. (1) The inclusion in (2.6) follows easily from the definitions and the fact that lim r 0 h(r) r t = ∞ for all t > 0. Next, in order to show (2.7), we must construct a continuous dimension function h satisfying condition (2.5) and a compact metric space X such that dim B (X) = 0 and U h (X) > 0. We construct the space X as follows. For a positive integer n, write I n = {0, 2(n + 1) − 1}, and for i ∈ I n define S n,i : the set X n is the union of the 2 n disjoint closed intervals I i1...in each with length equal to 1 2 n (n+1)! , and the sets X n are constructed inductively as follows: let X 0 = [0, 1] and for n = 1, 2, . . ., the set X n is obtained by deleting the middle n n+1 'th part of each of the intervals I i1...in−1 in X n−1 .
We first show that dim B (X) = 0. Indeed, if 1 2 n (n+1)! < r ≤ 1 2 (n−1) n! , then X can be covered by 2 n closed intervals with diameter equal to r and so It is clear that (2.5) is satisfied. We now show that H h (X) > 0. Let λ i1...in denote the Lebesgue measure restricted to the interval I i1...in and normalised so that λ i1...in (I i1...in ) = 1. Next, define the probability measure μ n by μ n = 1 2 n i1∈I1,...,in∈In λ i1...in . It is not difficult to see that there is a probability measure μ such that μ n converges weakly to μ. We now show that there is a constant c > 0 such that (2.9) We now prove the following claim.  Combining (2.9) and (2.10) we deduce that provided diam(U ) < r N . This proves inequality (2.8). Finally, it follows from (2.8) and the mass distribution principle that H h (X) ≥ 1 > 0.

Proofs of Theorems 2.2 and 2.3: Preliminary results
In this section we collect some basic notation and present several technical auxiliary lemmas that will be used in Sects. 4-6. We first list some useful properties of the covering number N r (X) and the packing number M r (X); recall that the covering number N r (X) and the packing number M r (X) of a metric space X are defined in (2.1).
Next, we list some useful properties of the Hewitt-Stromberg measures U h and V h ; recall that the Hewitt-Stromberg measures U h and V h are defined in Section 2.3.

Proposition 3.2. Let h be a continuous dimension function.
(1) For all metric spaces X and all E ⊆ X, we have U h (E) = U h (E).
(2) For all metric spaces X and all E ⊆ X, Proof. Let X be a metric space and E ⊆ X. It is clear that We first prove the following claim.
for all r > 0. Proof of Claim 1. Let d denote the metric in X. Recall (see Section 2.2) that we use the following notation, namely, if x ∈ X and r > 0, then C(x, r) denotes the closed ball with radius equal to r and centre at x, i.e. C(x, r) = {y ∈ X | d(x, y) ≤ r}. We now turn towards the proof of Claim 1. Let r > 0. Since h is continuous, we can choose a real number δ(r) with 0 < δ(r) ≤ 1 2 such that It follows from the definition of the packing number M r ( E ) that we can find a family ( C(x i , r) ) is a family of closed balls with y i ∈ E and d( and it therefore follows from Claim 1 that and Finally, letting ε tend to 0 in (3.5) and (3.6) gives the desired result.

Proposition 3.3. Let h be a continuous dimension function. Let X be a complete metric space and let C be a compact subset of
Finally, using (3.7) and taking the infimum over all countable families ( Vol. 92 (2018) On the Hausdorff and packing measures 723 (2) The proof of this statement is identical to the proof of the statement in Part (1) and is therefore omitted.

Proof of Theorem 2.3
The purpose of this section is to prove Theorem 2.3. For a dimension function h, we define the set H h by H h = X ∈ K GH for all t > 0 there is a positive integer N and

Proposition 4.1. Let h be a dimension function. Then the set H h is dense in
Proof. Let X ∈ K GH and let ρ > 0. Also, let d X denote the metric in X. We must now find a compact metric space Y ∈ K GH such that d GH (X, Y ) < ρ and Y ∈ H h . Since X is compact we can choose a finite subset E of X such that d H (X, E) < ρ 2 . Next, define the dimension function l : (0, ∞) → (0, ∞) by l(r) = rh(r), and note that it follows from [12,Theorem 36] that there is a compact metric space (Z, d Z ) such that Let μ denote the l-dimensional Hausdorff measure restricted to Z, and write Z 0 for the support of μ, i.e. Z 0 = supp μ. Next, we fix z 0 ∈ Z 0 and put z ) ) for x , x ∈ X and z , z ∈ K. It is clear that Y is compact, and so Y ∈ K GH . Below we show that d GH (X, Y ) < ρ and Y ∈ H h . This is the contents of the two claims below.
It is clear that f and g are isometries and we therefore conclude that This completes the proof of Claim 1.
Proof of Claim 2. Let t > 0. It follows from the compactness of K that we can choose finitely many points z 1 , . . . , Below we show that the statements in (4.4)-(4.6) are satisfied.

t); this proves (4.4).
It is also clear that C i,j ⊆ B(y i,j , t) for all i, j; this proves (4.5). Finally, we prove (4.6). We first show that In particular, we conclude that U j is an open subset of Z with U j ∩ Z 0 = ∅, and since Z 0 is the support of the l-dimensional Hausdorff measure restricted to Z, Vol. 92 (2018) On the Hausdorff and packing measures 725 Using Lemma 3.1 we deduce that for δ > 0, we have Also observe that it follows from the definition of the covering number N δ (K j ) that we can find a family B δ (K j ) of N δ (K j ) closed balls in Z with centres in K j and radii equal to δ that covers K j . In particular, diam(C) ≤ 2δ for all C ∈ B δ (K j ), and so Combining (4.8) and (4.9) now shows that (diam(C)) . (4.10) However, we conclude from (4.7) that C∈B δ (Kj ) l(diam(C)) ≥ 1 2 H l (K j ) for all 0 < δ ≤ δ j , and it therefore follows from (4.10) that This completes the proof of (4.6).
It follows immediately from (4.4)-(4.6) that Y ∈ H h . This completes the proof of Claim 2.
Finally, it follows from Claim 1 and Claim 2 that H h is dense in K GH .

Proposition 4.2. Let h be a continuous dimension function.
(1) The set Proof. (1) Using Proposition 4.1, it clearly suffices to show that (4.12) We will now prove (4.12). Let X ∈ H h . In order to prove (4.12), we must now show that U h (U ) = ∞ for all open subsets U of X with U = ∅. We therefore let U be an open subset of X with U = ∅, and proceed to show that Since U is non-empty and open there is x 0 ∈ U and t 0 > 0 with B X (x 0 , t 0 ) ⊆ U . Next, since X ∈ H h , we conclude that there is a positive integer N and (2) Using Part 1, it clearly suffices to prove that (4.13) We will now prove (4.13). Let X ∈ K GH and assume that U h (U ) = ∞ for all open subsets U of X with U = ∅. In order to prove (4.13), we must now show that U h (U ) = ∞ for all open subsets U of X with U = ∅. We therefore fix an open subset U of X with U = ∅, and proceed to show that U 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, this implies that if we write C = B(x, r 2 ), then C is compact and C ⊆ B(x, r) ⊆ U . Next, we prove the following claim. (4.14) However, since the set B(z, δ) is open and non-empty, it follows from the assumption about X that U h (B(z, δ)) = ∞, and we therefore conclude from (4.14) that U h (C ∩ V ) = ∞. This completes the proof of Claim 1. Finally, it follows immediately from Claim 1 and Proposition 3.3 that U h (C) = ∞, and since C ⊆ U , this implies that U h (U ) = ∞.
We can now prove Theorem 2.3.

Proof of Theorem 2.2.(1)
The purpose of this section is to prove Theorem 2.2.(1). For a dimension function h and r, c > 0, write Proof. This follows immediately from Lemma 3.1.

Proposition 5.2. Let h be a dimension function.
(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 s ) s∈Q + of open and dense subsets G s of K GH such that ∩ s∈Q + G s ⊆ T c . For s ∈ Q + , we define the set G s by We now prove that the sets G s are open and dense subsets of K GH such that ∩ s∈Q + G s ⊆ T c ; this is the contents of the three claims below.
Proof of Claim 1. This follows immediately from Lemma 5.1. This completes the proof of Claim 1.
Proof of Claim 2. Indeed, it is clear that {X ∈ K GH | X is finite} is dense in K GH , and since it is not difficult to see that {X ∈ K GH | X is finite} ⊆ ∪ 0<r<s L h r,c = G s , we therefore conclude that G s is dense in K GH . This completes the proof of Claim 2.
Since X ∈ ∩ s∈Q + G s ⊆ ∩ n G 1 n , we conclude that for each positive integer n, we can find r n < 1 n such that X ∈ L h rn,c , whence M rn (X) h(2r n ) < c. It follows immediately from this that U h (X) = lim inf r 0 M r (X) h(2r) ≤ lim inf n M rn (X) h(2r n ) ≤ c, and so X ∈ T c . This completes the proof of Claim 3.

Proof of Theorem 2.2.(2)
The purpose of this section is to prove Theorem 2.2.(2). 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 r, t, c > 0, write and L h r,t,c = X ∈ K GH there is a positive integer N and Also recall that for a dimension function h, the set H h is defined in (4.1). (1) For all X ∈ K GH , we have lim inf r 0 N r (X) h(2r) ≥ Uh(X).
(2) This statement follows immediately from Part (1). Proof. Let X ∈ L h r,t,c and let d X denote the metric in X. Also, in order to distinguish balls in different metric spaces, we will denote the open ball in X with radius equal to δ and centre at x ∈ X by B X (x, δ) 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 there is a positive integer N and x 1 , . . . , x N ∈ X , C 1 , . . . , C N ⊆ X , r 1 , . . . , r N ∈ (0, r) , such that x, x i ) and note that Φ is continuous. Since X is compact, we therefore conclude that there is x 0 ∈ X such that Φ(x 0 ) = sup x∈X Φ(x). For brevity write t 0 = Φ(x 0 ) = sup x∈X Φ(x), and note that since Also, since C i is compact and C i ⊆ B(x i , t), we conclude that For brevity write Finally, since C i ∈ Λ h ri,c and Λ h ri,c is open (by Lemma 6.2), we conclude that there is a positive real number ρ i > 0 with dN  16 ) . It follows from (6.1) and (6.2) that ρ > 0. We will now prove that Let Y ∈ B(X, ρ) and 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 metric space 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), i.e. B Z (z, δ) = {z ∈ X | d Z (z, z ) < δ}.
We must now show that Y ∈ L h r,t,c . Since d H (X, Y ) < ρ, we conclude that for each i, there is a point y i ∈ Y with d Z (x i , y i ) < ρ. Next, put It is clear that y 1 , . . . , y N ∈ Y , K 1 , . . . , K N ⊆ Y , r 1 , . . . , r N ∈ (0, r) .
In order to prove that Y ∈ L h r,t,c , it suffices to show that Y = ∪ i B Y (y i , t) , (6.5) K i ⊆ B Y (y i , t) for all i , (6.6) The proofs of (6.5)-(6.7) are the contents of the three claims below.
Proof of Claim 1. It is clear that ∪ i B Y (y i , t) ⊆ Y . In order to prove the reverse inclusion, we let y ∈ Y . Since d H (X, Y ) < ρ, we conclude that there is a point x ∈ X with d Z (x, y) < ρ. Also, since min i d X (x, x i ) = Φ(x) ≤ t 0 , we deduce that there is an index j with d X (x, x j ) ≤ t 0 . Finally, it follows from the definition of y j that d Z (x j , y j ) < ρ. Hence d Y (y, y j ) = d Z (y, y j ) ≤ 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 ) < ρ+t 0 +ρ = 2ρ + t 0 ≤ t, and so y ∈ B Y (y j , t) ⊆ ∪ i B Y (y i , t). This completes the proof of Claim 1.
Proof of Claim 2. Since C i ⊆ B X (x i , t), it follows from the definition of the numbers t i = inf{s | C i ⊆ B(x i , s)} and d i = t − t i , that Finally, combining (6.8) and (6.9) shows that We can now prove that K i ⊆ B Y (y i , t). Let y ∈ K i . Since y ∈ K i , we have dist(y, C i ) ≤ ρ ≤ di 16 < di 8 , and it therefore follows that there is x ∈ C i with d Z (x, y) ≤ di 8 . Also, we deduce from (6.10) that and so y ∈ B Y (y i , t). This completes the proof of Claim 2.
Claim 3. K i ∈ Λ h ri,c for all i. 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 sup x∈Ci dist(x, K i ) ≤ ρ . (6.11) Indeed, let x ∈ C i . Since d H (X, Y ) < ρ, we conclude that there is y ∈ Y such that d Z (x, y) < ρ. 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 the supremum over all x ∈ C i shows that sup x∈Ci dist(x, K i ) ≤ ρ. This completes the proof of (6.11). Next, we prove that sup y∈Ki dist(y, C i ) ≤ ρ . (6.12) 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 the supremum over all y ∈ K i shows that sup y∈Di dist(y, C i ) ≤ ρ. This completes the proof of (6.12).