FINE VARIATION AND FRACTAL MEASURES

G. A. Edgar

The Ohio State University

revised October, 1994

Abstract. Thomson noted that (in the line) the Hausdorff measures can be considered to be fine variations for appropriate choices of derivation basis and set function. We show that this point of view remains interesting in a general separable metric space. Use of the “centered ball” basis yields an alternate description of the covering measures of Saint Raymond and Tricot. Use of a “closed set” basis yields the Hausdorff measures. This paper may be considered a counterpart of [7], where the corresponding study of the packing measure may be found.

0. Introduction

The best-known fractal measures are the Hausdorff measures ℋ^{s} for s > 0. Saint Raymond and Tricot [12] defined a variant, known as the (centered ball) covering measures 𝒞^{s}. They differ by at most a constant factor from the Hausdorff measures, so they may be used in the computation of the Hausdorff dimension.

Thomson [13] studied “variation” measures defined on metric spaces. To a setfunction, such as the s-th power of the diameter, he associated a “full” variation (or Method III measure) and a “fine” variation (or Method IV measure). [The term “method III” has been used for other constructions as well, so we do not use it here.] For the interval basis on the line, the full variation is related to the packing measure, and the fine variation is the Hausdorff measure. Meinershagen [10] showed that packing measure is exactly the full variation associated with the balanced-interval basis. This characterization was extended to general metric spaces in [7].

In this paper we show that the St.-Raymond-Tricot covering measure is a fine variation when we use the closed centered-ball basis. Corresponding integral and density results are proved for this setting. This paper may be considered a counterpart of [7], where the corresponding study of the packing measure may be found, but the two papers may be read independently.

The Hausdorff measures are treated in a similar way (with the same proofs in most cases) by changing to a different derivation basis in §6. Covering measure

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 28A80, secondary 26A39.

Key words and phrases. fractal measure, covering measure, Hausdorff measure, fine variation, derivation basis.

defined using the diameter of a ball (rather than the radius) is discussed in §4. The upper density defined with diameter may be (in some metric spaces) a nonmeasurable function (Example 5.3).

Background material on Carathéodory’s outer measures and metric outer measures may be found in [4, § 1.3] or [5, Chapt. 5]; but an even better reference is Carathéodory’s original paper [cara].

1. Covering measure

We begin with the definition of the covering measure (see [12]). It is a slight variant of the Hausdorff measure. Let (S, ρ) be a separable metric space. For x ∈ S and r > 0, the open ball is

B_{r}(x) = { y ∈ S : ρ(y, x) < r } ,

and the closed ball is

\overline{B}_{r}(x) = \{ y ∈ S : ρ(y, x) ≤ r \} .

The diameter of a set E ⊆ S is

diam E = sup { ρ(x, y) : x, y ∈ E } .

Let A ⊆ S be a set. A centered-ball cover of A is a countable collection \{\overline{B}_{r_{1}}(x_{1}), \overline{B}_{r_{2}}(x_{2}), ⋯ \} of closed balls with center in A that covers A:

x_{i} ∈ A, A ⊆ ⋃_{i} \overline{B}_{r_{i}}(x_{i}).

A centered-ball packing of A is a countable disjoint collection

\{\overline{B}_{r_{1}}(x_{1}), \overline{B}_{r_{2}}(x_{2}), ⋯ \}

of closed balls with center in A:

x_{i} ∈ A, \overline{B}_{r_{i}}(x_{i}) ∩ \overline{B}_{r_{j}}(x_{j}) = ∅ for i ≠ j.

Let μ be a finite Borel measure on the metric space S. For any x ∈ S, the upper s-density of μ at x is

\overline{D}_{μ}^{s}(x) = lim sup_{r→0} \frac{μ(\overline{B}_{r}(x))}{(2r)^{s}}.

If x_{0} is an isolated point, then \overline{B}_{r}(x_{0}) = \{x_{0}\} for r small enough. So of course we define \overline{D}_{μ}^{s}(x_{0}) = ∞ if μ(\{x_{0}\}) > 0. Let us say by convention that \overline{D}_{μ}^{s}(x_{0}) = 0 if μ({x_{0}}) = 0.

(1.1) Proposition. Let μ be a finite Borel measure and let s > 0. Then the function \overline{D}_{μ}^{s} is a Borel function.

Proof. First, we note that for fixed r, the map x ↦ μ(\overline{B}_{r}(x)) is a Borel function. Indeed, for given t > 0, we claim that the set

V = \{ x ∈ S : μ(\overline{B}_{r}(x)) < t \}

is an open set. Let x_{0} ∈ V. That is, μ(\overline{B}_{r}(x_{0})) < t. The balls \overline{B}_{r+1/n}(x_{0}) decrease to \overline{B}_{r}(x_{0})), so there is n with μ(\overline{B}_{r+1/n}(x_{0})) < t. Now for any x with ρ(x, x_{0}) < 1/n, the ball \overline{B}_{r}(x) is contained in \overline{B}_{r+1/n}(x_{0}) so μ(\overline{B}_{r}(x)) < t. Thus V is an open set.

Next, for fixed x, the map r ↦ μ(\overline{B}_{r}(x)) is right-continuous. Indeed,

\overline{B}_{r_{0}}(x) = ⋂_{r>r_{0}} \overline{B}_{r}(x),

so μ(\overline{B}_{r}(x)) → μ(\overline{B}_{r_{0}}(x)) as r ↓ r_{0}.

The denominator (2r)^{s} is also a right-continuous function of r. So in the definition of \overline{D}_{μ}^{s}, the lim sup may be computed using only rational values:

\overline{D}_{μ}^{s}(x) = lim_{n→∞} sup \{ \frac{μ(\overline{B}_{r}(x))}{(2r)^{s}} : r ∈ Q, 0 < r < \frac{1}{n} \} .

Therefore it is also a Borel function. □

(1.2) Definition. Let s be a positive number. For ε > 0, define

𝒞_{ε}^{s}(A) = inf ∑_{i}(2r_{i})^{s},

where the infimum is over all countable covers \{\overline{B}_{r_{i}}(x_{i})\} of A by centered closed balls with r_{i} ≤ ε. [Of course, in Euclidean space diam \overline{B}_{r}(x) = 2r, but in a general separable metric space this need not be true.] The s-dimensional covering premeasure of A is

𝒞̃^{s}(A) = lim_{ε→0} 𝒞_{ε}^{s}(A).

The s-dimensional covering outer measure is the outer measure 𝒞^{s} defined by making 𝒞̃^{s} increasing:

𝒞^{s}(A) = sup { 𝒞̃^{s}(E) : E ⊆ A } .

Then 𝒞^{s} is a metric outer measure on S. ([12, Lemma 3.1]; the result remains correct in any separable metric space S.) We will restrict attention here to separable metric spaces, since 𝒞^{s}(A) = ∞ for any non-separable set.

The definition is a bit awkward to use, because of the “increasing” step added on the end. But the set function 𝒞̃^{s} is not increasing so this final step is needed.

Here are a few observations about the definition. The proofs in [12] often apply unchanged for a general separable metric space. (1.3) The covering measure 𝒞^{s} and the Hausdorff measure ℋ^{s} differ at most by

a constant factor:

2^{−s}𝒞^{s}(E) ≤ ℋ^{s}(E) ≤ 𝒞^{s}(E).

In particular, the values of s for which ℋ^{s}(E) = 0 are the same as those for which 𝒞^{s}(E) = 0; and similarly for ℋ^{s}(E) = ∞ and 𝒞^{s}(E) = ∞. [12, Lemma 3.3]

(1.4) The value of 𝒞^{s}(E) is unchanged if we use covers by open balls rather than

covers by closed balls. (1.5) In Euclidean space, if μ is a finite Borel measure, and E is a Borel set with

𝒞^{s}(E) < ∞, then

𝒞^{s}(E) inf_{x∈E} \overline{D}_{μ}^{s}(x) ≤ μ(E) ≤ 𝒞^{s}(E) sup_{x∈E} \overline{D}_{μ}^{s}(x).

[12, 1.1]; we will see a strengthened version of this below. (1.6) Is 𝒞^{s} a regular outer measure? That is, for any set A, there is a measurable

set E ⊇ A with 𝒞^{s}(E) = 𝒞^{s}(A). I do not know the answer.

2. Fine variation

We now consider (a special case of) Thomson’s fine variation [13], [14]. The variations may be defined for a general “derivation basis”. For the moment we will use only the “centered ball” basis, so reference to the basis will be suppressed from our notation and terminology.

Let (S, ρ) be a separable metric space. A constituent is a pair (r, x), where x ∈ S is a point, and r is a positive real number. [The constituent (r, x) represents the closed ball \overline{B}_{r}(x). In a general metric space, different centers x, x′ and/or different radii r, r′ may represent identical point sets:

\overline{B}_{r}(x) = \overline{B}_{r′}(x′),

so we emphasize a center and radius are given as the constituent.]

A packing is a disjoint collection π of constituents: that is, \overline{B}_{r}(x)∩\overline{B}_{r′}(x′) = ∅ if (r, x), (r′, x′) ∈ π, (r, x) ≠ (r′, x′). A packing of a set A is a packing π such that x ∈ A for all (r, x) ∈ π. Note that since S is separable, any packing by balls of positive radius must be a countable packing. A centered cover of a set A is a collection β of constituents (r, x) with x ∈ A and A ⊆ ⋃_{(r,x)∈β} \overline{B}_{r}(x). A fine cover (or Vitali cover) of a set A is a centered cover β of A such that, for every x ∈ A and every ε > 0, there is r < ε with (r, x) ∈ β.

We will need the following Vitali-type theorem. The usual (Banach) proof of Vitali’s theorem proves this without trouble (for example [4, Theorem 6.2.1], [8, Theorem 1.10]). But note the use of closed balls.

(2.1) Theorem. Let (S, ρ) be a separable metric space, let E ⊆ S be a set, let β be a fine cover of E, and let s > 0. Then there is a (finite or infinite) packing π = {(r_{i}, x_{i})} ⊆ β such that:

either  ∑_{i}(2r_{i})^{s} = ∞  or  𝒞^{s} (E \ ⋃_{i} \overline{B}_{r_{i}}(x_{i})) = 0.

(2.2) Definition. Let h be a finite nonnegative constituent function: that is, for each constituent (r, x), let 0 ≤ h(r, x) < ∞. If β is a cover, write

V_{β}(h) = sup ∑_{(r,x)∈π} h(r, x),

where the supremum is over all packings π ⊆ β. The fine variation of h is

V_{∗}(h) = inf V_{β}(h),

where the infimum is over all fine covers β.

The intersection of two fine covers may not be a fine cover at all, so this infimum is not a “limit” in the sense of Moore-Smith. If h(r, x) = 0 whenever x is outside a set A, then of course it is enough to use only fine covers β of A in the infimum.

When the constituent function h is of the special form h(r, x) = f(x)(2r)^{s}, for some nonnegative point-function f: S → IR, we will write V_{β}^{s}(f) = V_{β}(h) and V_{∗}^{s}(f) = V_{∗}(h). We will call V_{∗}^{s}(f) the fine s-variation of f.

The following is a special case of Thomson’s general results on fine variations [13]. We write 1l_{A} for the indicator function of a set A, and h 1l_{A} for the constituent function

(h 1l_{A})(r, x) = h(r, x) 1l_{A}(x).

(2.3) Theorem. Let h be a nonnegative constituent function. Then μ defined by μ(A) = V_{∗}(h 1l_{A}) for all A ⊆ S is a metric outer measure on S.

If S is a separable metric space, then the Cantor-Bendixson Theorem [9, p. 253] states that S may be written as a disjoint union

S = S_{0} ∪ Q,

where Q is countable, and S_{0} has no isolated points. (Q consists of all points x_{0} such that \overline{B}_{r}(x_{0}) is countable, for some r > 0.) If a constituent function h is continuous in the sense that

lim_{r→0} h(r, x) = 0

for every x, then the measure V_{∗}(h 1l_{A}) vanishes on single points, so it vanishes on countable sets like Q. This fact will be used to reduce many of our assertions to the case where the metric space S has no isolated points.

3. Covering measure and fine variation

Next is the main result. The fine variation measure defined by the set function h(r, x) = (2r)^{s} coincides with the covering measure. The one-dimensional version (for Hausdorff measure) occurs in [14, Theorem 6.4].

(3.1) Theorem. Let (S, ρ) be a separable metric space, and let s > 0. Then for every set E ⊆ S, we have V_{∗}^{s}(1l_{E}) = 𝒞^{s}(E).

Proof. (a) We first prove 𝒞^{s}(E) ≤ V_{∗}^{s}(1l_{E}). If V_{∗}^{s}(1l_{E}) = ∞, there is nothing to prove, so suppose V_{∗}^{s}(1l_{E}) < ∞. Let β be a fine cover of E with V_{β}^{s}(1l_{E}) < ∞. Let ε > 0. By the Vitali theorem (2.1), there is a packing {(r_{i}, x_{i})} ⊆ β with r_{i} < ε and either ∑(2r_{i})^{s} = ∞ or 𝒞^{s}(E \ ⋃ \overline{B}_{r_{i}}(x_{i})) = 0. But ∑(2r_{i})^{s} ≤ V_{β}^{s}(1l_{E}) < ∞, so 𝒞^{s}(E \ ⋃ \overline{B}_{r_{i}}(x_{i})) = 0. Then 𝒞_{ε}^{s}(E \ ⋃ \overline{B}_{r_{i}}(x_{i})) = 0 for all ε > 0. Thus

𝒞_{ε}^{s}(E) ≤ 𝒞_{ε}^{s} (⋃_{i=1}^{∞} \overline{B}_{r_{i}}(x_{i})) ≤ ∑_{i=1}^{∞}(2r_{i})^{s} ≤ V_{β}^{s}(1l_{E}).

Now let ε → 0 to obtain 𝒞̃^{s}(E) ≤ V_{β}^{s}(1l_{E}). Then take the infimum over β to obtain 𝒞̃^{s}(E) ≤ V_{∗}^{s}(1l_{E}). Finally, take the supremum of this inequality over all subsets, to obtain 𝒞^{s}(E) ≤ V_{∗}^{s}(1l_{E}).

(b) Next we prove: if 𝒞^{s}(E) = 0, then V_{∗}^{s}(1l_{E}) = 0. Since S is separable, and V_{∗}^{s}(1l_{{x}}) = 0 for any single point x, we may reduce to the case where E contains no isolated points.

Let ε > 0. For each integer n, since 𝒞_{1/n}^{s}(E) = 0, there is a centered cover

{ (r_{in}, x_{in}) : i ∈ IN }

of E with r_{in} < 1/n, x_{in} ∈ E, and

∑_{i} (2r_{in})^{s} < \frac{ε}{2^{n+1}} .

Now for each i and n let

β_{in} = { (r_{in}, y) : y ∈ E, ρ(y, x_{in}) ≤ r_{in} } .

Then

β = ⋃_{i,n} β_{in}

is a fine cover of E. Let π ⊆ β be a packing. For each i, n, because all elements of β_{in} contain the point x_{in}, there is at most one element of β_{in} in π. Thus

∑_{(r,x)∈π}(2r)^{s} ≤ ∑_{i,n}(2r_{in})^{s} ≤ ∑_{n} \frac{ε}{2^{n+1}} = ε.

Thus V_{β}^{s}(1l_{E}) ≤ ε. So V_{∗}^{s}(1l_{E}) ≤ ε. But ε > 0 was arbitrary, so V_{∗}^{s}(1l_{E}) = 0.

(c) Finally, we prove that V_{∗}^{s}(1l_{E}) ≤ 𝒞^{s}(E). Again, we may reduce to the case where E contains no isolated points of S. If 𝒞^{s}(E) = ∞, there is nothing to prove, so assume 𝒞^{s}(E) < ∞. Let μ be the restriction of 𝒞^{s} to E: that is, μ(A) = 𝒞^{s}(A∩E) for all A. Then μ is a finite metric outer measure.

We will decompose E using the upper s-density \overline{D}_{μ}^{s}. Fix a number α > 1. Write

E_{1} = \{ x ∈ E : \overline{D}_{μ}^{s}(x) ≤ α^{−3} \}
E_{2} = \{ x ∈ E : \overline{D}_{μ}^{s}(x) > α^{−3} \} .

Consider first E_{1}. For n ∈ IN, write

F_{n} = \{ x ∈ E_{1} : \frac{μ(\overline{B}_{r}(x))}{(2r)^{s}} < α^{−2} for all r < \frac{1}{n} \} .

Then F_{n} increases to E_{1} as n → ∞, since α^{−2} > α^{−3}.

I claim that 𝒞^{s}(F_{n}) = 0. If ε < 1/n, then when F_{n} is covered by \{\overline{B}_{r_{i}}(x_{i})\} with r_{i} < ε, we have

∑(2r_{i})^{s} ≥ α^{2} ∑ μ(\overline{B}_{r_{i}}(x_{i})) ≥ α^{2}μ (⋃ \overline{B}_{r_{i}}(x_{i})) ≥ α^{2}μ(F_{n}) = α^{2}𝒞^{s}(F_{n}).

Therefore 𝒞_{ε}^{s}(F_{n}) ≥ α^{2}𝒞^{s}(F_{n}). Let ε → 0 to obtain 𝒞̃^{s}(F_{n}) ≥ α^{2}𝒞^{s}(F_{n}). Therefore 𝒞^{s}(F_{n}) ≥ α^{2}𝒞^{s}(F_{n}). Now 𝒞^{s}(F_{n}) < ∞ and α^{2} > 1, so 𝒞^{s}(F_{n}) = 0.

Thus 𝒞^{s}(F_{n}) = 0 for all n. By countable subadditivity, we conclude 𝒞^{s}(E_{1}) = 0. By part (b), V_{∗}^{s}(1l_{E_{1}}) = 0 as well.

Next consider the set E_{2}. Since α^{−4} < α^{−3}, the set

β = \{ (r, x) constituent : x ∈ E_{2}, \frac{μ(\overline{B}_{r}(x))}{(2r)^{s}} > α^{−4} \}

is a fine cover of E_{2}. Now if π ⊆ β is a packing, then

∑_{(r,x)∈π}(2r)^{s} < α^{4} ∑_{π} 𝒞^{s}(\overline{B}_{r}(x) ∩ E) ≤ α^{4}𝒞^{s}(E).

This is true for all π ⊆ β, so V_{β}^{s}(1l_{E_{2}}) ≤ α^{4}𝒞^{s}(E), and thus V_{∗}^{s}(1l_{E_{2}}) ≤ α^{4}𝒞^{s}(E). Combining the two parts, we have

V_{∗}^{s}(1l_{E}) ≤ V_{∗}^{s}(1l_{E_{1}}) + V_{∗}^{s}(1l_{E_{2}}) ≤ 0 + α^{4}𝒞^{s}(E).

Take the infimum over all α > 1 to obtain V_{∗}^{s}(1l_{E}) ≤ 𝒞^{s}(E). □

For special h of the form f(x)(2r)^{s}, the fine variation V_{∗}^{s}(f) is the integral with respect to the covering measure 𝒞^{s}. Note that f is not permitted to have the value ∞.

(3.2) Proposition. Let S be a separable metric space, let s > 0, and let f be a nonnegative real-valued Borel function on S. Then

V_{∗}^{s}(f) = ∫_{S} f(x) 𝒞^{s}(dx).

Proof. First, by Theorem 3.1, we have V_{∗}^{s}(1l_{A}) = 𝒞^{s}(A) for any set A. Clearly, if a ≥ 0 is a constant and f ≥ 0 is a finite non-negative function, then V_{∗}^{s}(af) = aV_{∗}^{s}(f).

Now both 𝒞^{s}(A) and V_{∗}^{s}(f 1l_{A}) are metric outer measures, so Borel sets are measurable. If A_{n} are disjoint Borel sets and a_{n} ≥ 0 are constants, then the simple function f = ∑ a_{n}1l_{A_{n}} satisfies V_{∗}^{s}(f) = ∫_{S} f(x) 𝒞^{s}(dx).

Finally, if f is a nonnegative Borel function, then there is a sequence f_{n} of nonnegative Borel measurable simple functions that increases to f. If c < 1, then the sets

E_{n} = { x : f_{n}(x) ≥ cf(x) }

increase to S. But V_{∗}^{s}(f_{n}) ≥ cV_{∗}^{s}(f 1l_{E_{n}}), so lim_{n} V_{∗}^{s}(f_{n}) ≥ cV_{∗}^{s}(f). Let c → 1 to conclude that V_{∗}^{s}(f_{n}) → V_{∗}^{s}(f). Therefore we have

V_{∗}^{s}(f) = lim_{n} V_{∗}^{s}(f_{n}) = lim_{n} ∫_{S} f_{n}(x) 𝒞^{s}(dx) = ∫_{S} f(x) 𝒞^{s}(dx),

as required. □

Next is a strengthened form of (1.5). It identifies \overline{D}_{μ}^{s} as the Radon-Nikodym derivative of μ with respect to 𝒞^{s}. Because 𝒞^{s} is not σ-finite, the best we can hope for is equality for sets E on which 𝒞^{s} is finite (or σ-finite). We must also rule out \overline{D}_{μ}^{s}(x) = ∞.

(3.3) Theorem. (a) Let μ be a finite Borel measure on the separable metric space S and let E ⊆ S be a Borel set. Then

μ(E) ≥ ∫_{E} \overline{D}_{μ}^{s}(x) 𝒞^{s}(dx).

(b) If, in addition, 𝒞^{s}(E) < ∞, and \overline{D}_{μ}^{s} < ∞ on E, then

μ(E) = ∫_{E} \overline{D}_{μ}^{s}(x) 𝒞^{s}(dx).

Proof. (a) Write E = E_{0} ∪ Q, where Q is countable, and E_{0} has no isolated points. Then 𝒞^{s}(Q) = 0, so ∫_{Q} \overline{D}_{μ}^{s}(x) 𝒞^{s}(dx) = 0. Thus we may reduce to the case where E contains no isolated points.

Let U ⊇ E be an open set. Let f be a finite Borel function, 0 ≤ f ≤ \overline{D}_{μ}^{s} with strict inequality f(x) < \overline{D}_{μ}^{s}(x) whenever \overline{D}_{μ}^{s}(x) > 0. Now

β = \{ (r, x) constituent : x ∈ E, \overline{B}_{r}(x) ⊆ U, \frac{μ(\overline{B}_{r}(x))}{(2r)^{s}} ≥ f(x) \}

is a fine cover of E. If π ⊆ β is a packing, then

∑_{(r,x)∈π} f(x)(2r)^{s} ≤ ∑_{π} μ(\overline{B}_{r}(x)) = μ (⋃_{π} \overline{B}_{r}(x)) ≤ μ(U).

So V_{β}^{s}(f 1l_{E}) ≤ μ(U), and therefore V_{∗}^{s}(f 1l_{E}) ≤ μ(U). Taking the infimum over U, we obtain V_{∗}^{s}(f 1l_{E}) ≤ μ(E). Now f < ∞ on E, so this means ∫_{E} f(x)𝒞^{s}(dx) ≤ μ(E). But then by the choice of f we may conclude ∫_{E} \overline{D}_{μ}^{s}(x)𝒞^{s}(dx) ≤ μ(E).

(b) Now suppose 𝒞^{s}(E) < ∞, and \overline{D}_{μ}^{s} < ∞ on E. We claim that

∫_{E} \overline{D}_{μ}^{s}(x)𝒞^{s}(dx) ≥ μ(E).

Since \overline{D}_{μ}^{s} < ∞, we must have μ(\{x\}) = 0 for all x ∈ E. So again we may reduce to the case where E contains no isolated points of S.

We claim first that μ ≪ 𝒞^{s} on E. Let F ⊆ E with 𝒞^{s}(F) = 0. We must show μ(F) = 0. Since 𝒞^{s}(F) = 0, we have 𝒞_{ε}^{s}(F) = 0 for all ε > 0. Now for n ∈ IN, let

F_{n} = \{ x ∈ F : \frac{μ(B_{r}(x))}{(2r)^{s}} < n for all r < \frac{1}{n} \} .

So F_{n} increases to F since \overline{D}_{μ}^{s}(x) < ∞ on E. Now if ε < 1/n, then any centered ε-cover \{\overline{B}_{r_{i}}(x_{i})\} of F_{n} satisfies

∑(2r_{i})^{s} > \frac{1}{n} ∑ μ(\overline{B}_{r_{i}}(x_{i})) ≥ \frac{1}{n}μ(F_{n}).

Thus, 𝒞_{ε}^{s}(F_{n}) ≥ (1/n)μ(F_{n}). Therefore μ(F_{n}) = 0. By the countable subadditivity of μ, we conclude that μ(F) = 0.

Now let β be a fine cover of E. Then

β′ = \{ (r, x) ∈ β : \frac{μ(\overline{B}_{r}(x))}{(2r)^{s}} ≤ \overline{D}_{μ}^{s}(x) + ε \}

is also a fine cover of E. But by the Vitali theorem (2.1) there is a packing π ⊆ β′ such that either ∑_{π}(2r)^{s} = ∞ or 𝒞^{s}(E\⋃_{π} \overline{B}_{r}(x)) = 0 so that μ(E\⋃_{π} \overline{B}_{r}(x)) = 0. In either case

∑_{(r,x)∈π} (\overline{D}_{μ}^{s}(x) + ε)(2r)^{s} ≥ μ(E) :

This is true in the first case since the left-hand side is ∞, and in the second case since ∑(\overline{D}_{μ}^{s}(x) + ε)(2r)^{s} ≥ ∑ μ(\overline{B}_{r}(x)) ≥ μ(E). Thus

V_{β}^{s}((\overline{D}_{μ}^{s} + ε) 1l_{E}) ≥ μ(E).

The infimum over all β yields V_{∗}^{s}((\overline{D}_{μ}^{s} + ε) 1l_{E}) ≥ μ(E). The integrand is finite, so this means ∫_{E} \overline{D}_{μ}^{s}(x)𝒞^{s}(dx) + ε𝒞^{s}(E) ≥ μ(E). Since 𝒞^{s}(E) < ∞, we may let ε → 0 to obtain ∫_{E} \overline{D}_{μ}^{s}(x)𝒞^{s}(dx) ≥ μ(E). □

(3.4) Corollary. Let S be a separable metric space, let s > 0, and let E ⊆ S be a Borel set with 𝒞^{s}(E) < ∞.

(a) Let μ be a finite Borel measure. When the products are well-defined [i.e. not 0 · ∞], we have

𝒞^{s}(E) inf_{x∈E} \overline{D}_{μ}^{s}(x) ≤ μ(E) ≤ 𝒞^{s}(E) sup_{x∈E} \overline{D}_{μ}^{s}(x).

(b) The density theorem: Let μ be the restriction of 𝒞^{s} to E. Then \overline{D}_{μ}^{s}(x) = 1 for 𝒞^{s}-almost every x ∈ E and \overline{D}_{μ}^{s}(x) = 0 for 𝒞^{s}-almost every x ∉ E.

Proof. (a) The constant inf_{x∈E} \overline{D}_{μ}^{s}(x) is a Borel function ≤ \overline{D}_{μ}^{s}. By Theorem 3.3,

𝒞^{s}(E) inf_{x∈E} \overline{D}_{μ}^{s}(x) ≤ ∫_{E} \overline{D}_{μ}^{s}(x)𝒞^{s}(dx) ≤ μ(E).

Similarly,

𝒞^{s}(E) sup_{x∈E} \overline{D}_{μ}^{s}(x) ≥ ∫_{E} \overline{D}_{μ}^{s}(x)𝒞^{s}(dx) ≥ μ(E).

(b) Let μ be the restriction of 𝒞^{s} to E: that is, μ(A) = 𝒞^{s}(A ∩ E) for all A. So μ is a finite Borel measure.

For c < 1, let E_{c} = \{ x ∈ E : \overline{D}_{μ}^{s}(x) ≤ c \}. Then

𝒞^{s}(E_{c}) = μ(E_{c}) ≤ 𝒞^{s}(E_{c}) sup_{x∈E_{c}} \overline{D}_{μ}^{s}(x) ≤ c𝒞^{s}(E_{c}).

So 𝒞^{s}(E_{c}) = 0. Thus \overline{D}_{μ}^{s}(x) ≥ 1 a.e. on E.

For c > 1, let E^{c} = \{ x ∈ E : \overline{D}_{μ}^{s}(x) ≥ c \}. Then

𝒞^{s}(E^{c}) = μ(E^{c}) ≥ 𝒞^{s}(E^{c}) inf_{x∈E^{c}} \overline{D}_{μ}^{s}(x) ≥ c𝒞^{s}(E^{c}).

So 𝒞^{s}(E^{c}) = 0. Thus \overline{D}_{μ}^{s}(x) ≥ 1 a.e. on E.

For c > 0, let F_{c} = \{ x ∈ S \ E : \overline{D}_{μ}^{s}(x) ≥ c \}. Then

0 = μ(F_{c}) ≥ 𝒞^{s}(F_{c}) inf_{x∈F_{c}} \overline{D}_{μ}^{s}(x) ≥ c𝒞^{s}(F_{c}).

So 𝒞^{s}(F_{c}) = 0. Thus \overline{D}_{μ}^{s}(x) = 0 a.e. on S \ E. □

4. Diameter vs. radius

Now let us consider a possible variant of the definitions used here. [In fact, these are the definitions I used at first.] Rather than using (2r)^{s} for a closed ball \overline{B}_{r}(x), let us use the actual diameter ( diam \overline{B}_{r}(x))^{s}. In many metric spaces (such as Euclidean space) this makes no difference. But in other metric spaces (including certain subsets of Euclidean space) it does make a difference. In this section we will write the upper s-density of a measure μ as

\overline{Δ}_{μ}^{s}(x) = lim sup_{r→0} \frac{μ(\overline{B}_{r}(x))}{( diam \overline{B}_{r}(x))^{s}} .

In some metric spaces \overline{Δ}_{μ}^{s} may fail to be a measurable function (Example 5.3). For fixed r > 0, the function x ↦ μ(\overline{B}_{r}(x)) is a Borel function (in fact upper semicontinuous). But the function x ↦ diam \overline{B}_{r}(x) need not be a measurable function (Example 5.2). If the metric space satisfies lim_{r↓a} diam \overline{B}_{r}(x) = diam \overline{B}_{a}(x), then x ↦ diam \overline{B}_{r}(x) is measurable, and then \overline{Δ}_{μ}^{s} will be measurable. See [1, Lemma 3] and §7, below.

In this section we will temporarily use the notation 𝒯^{s} for the appropriate covering outer measure:

(4.1) Definition. Let s be a positive number. For ε > 0, define

𝒯_{ε}^{s}(A) = inf ∑_{i} ( diam \overline{B}_{r_{i}}(x_{i}))^{s},

where the infimum is over all countable covers of A by centered closed balls with diameter ≤ ε. The s-dimensional covering pre-measure of A is

𝒯̃^{s}(A) = lim_{ε→0} 𝒯_{ε}^{s}(A).

The s-dimensional covering outer measure is the outer measure 𝒯^{s} defined by making 𝒯̃^{s} increasing:

𝒯^{s}(A) = sup { 𝒯̃^{s}(E) : E ⊆ A } .

Then 𝒯^{s} is a metric outer measure on S.

The fine variations to be used are, of course, the same as before, but we will write W_{∗}(h) and now we will use the notation W_{∗}^{s}(f) when the constituent function h is of the special form h(r, x) = f(x)( diam \overline{B}_{r}(x))^{s}.

With the new notation, most of the same results are proved as before:

(4.2) Theorem. Let (S, ρ) be a separable metric space, and let s > 0. Then for every set E ⊆ S, we have W_{∗}^{s}(1l_{E}) = 𝒯^{s}(E).

Note that the sets F_{n} in the proof are not at first known to be measurable sets, but the argument shows they have outer measure zero, so they are, in fact, measurable.

(4.3) Proposition. Let S be a separable metric space, let s > 0, and let f be a nonnegative real-valued Borel function on S. Then

W_{∗}^{s}(f) = ∫_{S} f(x) 𝒯^{s}(dx).

If g is a non-negative real function on S, possibly not 𝒯^{s}-measurable, the integral ∫_{S} g(x)𝒯^{s}(dx) may not be defined. We will use the upper and lower integrals:

\overline{∫} g(x)𝒯^{s}(dx) = inf \{ ∫ f(x)𝒯^{s}(dx) : f ≥ g, f Borel measurable \} ,
\underline{∫} g(x)𝒯^{s}(dx) = sup \{ ∫ f(x)𝒯^{s}(dx) : f ≤ g, f Borel measurable \} .

(4.4) Theorem. (a) Let μ be a finite Borel measure on the separable metric space S and let E ⊆ S be a Borel set. Then

μ(E) ≥ \underline{∫} _{E}\overline{Δ}_{μ}^{s}(x) 𝒯^{s}(dx).

(b) If, in addition, 𝒯^{s}(E) < ∞, and \overline{Δ}_{μ}^{s} < ∞ on E, then

μ(E) ≤ \overline{∫} _{E}\overline{Δ}_{μ}^{s}(x) 𝒯^{s}(dx).

(c) If, in addition, \overline{Δ}_{μ}^{s} is Borel measurable, then

μ(E) = ∫_{E} \overline{Δ}_{μ}^{s}(x) 𝒯^{s}(dx).

(4.5) Corollary. Let S be a separable metric space, let s > 0, and let E ⊆ S be a Borel set with 𝒯^{s}(E) < ∞.

(a) Let μ be a finite measure. When the products are well-defined [i.e. not 0·∞], we have

𝒯^{s}(E) inf_{x∈E} \overline{Δ}_{μ}^{s}(x) ≤ μ(E) ≤ 𝒯^{s}(E) sup_{x∈E} \overline{Δ}_{μ}^{s}(x).

(b) The density theorem: Let μ be the restriction of 𝒯^{s} to E. Then \overline{Δ}_{μ}^{s}(x) = 1 for 𝒯^{s}-almost every x ∈ E and \overline{Δ}_{μ}^{s}(x) = 0 for 𝒯^{s}-almost every x ∉ E.

5. Irregular examples

Here we display three examples showing that regularity properties (for diameter) may fail in certain metric spaces:

(a) A set S ⊆ IR and a finite Borel measure μ on S so that

lim sup_{r→0} \frac{μ(B_{r}(0))}{( diam B_{r}(0))^{s}} ≠ lim sup_{r→0} \frac{μ(\overline{B}_{r}(0))}{( diam \overline{B}_{r}(0))^{s}} .

(b) A set S ⊆ IR^{2} so that the function x ↦ diam \overline{B}_{1}(x) is a non-measurable function.

(c) A (separable) metric space S and a finite Borel measure μ on S so that \overline{Δ}_{μ}^{1} is a non-measurable function.

(5.1) Example. Let s = 1 and α = 3/2. The set is

S = {0} ∪ ⋃_{n=1}^{∞}[2^{−n}, 2^{−n}α] ∪ ⋃_{n=1}^{∞}[−2^{−n}α, −2^{−n}).

Let ρ be the restriction of the usual metric of IR to the subset S. Let μ be a discrete measure with mass 2^{−n} at the point 2^{−n}, for n = 1, 2, · · ·. Certainly μ is a (regular) Borel measure on S with finite total mass.

Now in the metric space S, for 2^{−n} < r ≤ 2^{−n}α,

\frac{μ (B_{r}(0))}{diam B_{r}(0)} = \frac{μ (\overline{B}_{r}(0))}{diam \overline{B}_{r}(0)} = \frac{2^{−n+1}}{2r} < 1.

For r = 2^{−n},

\frac{μ (B_{r}(0))}{diam B_{r}(0)} = \frac{2^{−n}}{2^{−n}α} = \frac{1}{α} < 1,
\frac{μ (\overline{B}_{r}(0))}{diam \overline{B}_{r}(0)} = \frac{2^{−n+1}}{2^{−n} + 2^{−n−1}α} = \frac{4}{2 + α} = \frac{8}{7} > 1.

And for 2^{−n−1}α < r < 2^{−n},

\frac{μ (B_{r}(0))}{diam B_{r}(0)} = \frac{μ (\overline{B}_{r}(0))}{diam \overline{B}_{r}(0)} = \frac{2^{−n}}{2^{−n}α} = \frac{1}{α} < 1.

So we have

lim sup \frac{μ (\overline{B}_{r}(0))}{diam \overline{B}_{r}(0)} = \frac{8}{7}  but  lim sup \frac{μ (B_{r}(0))}{diam B_{r}(0)} ≤ 1.

(5.2) Example. Let V be a non-measurable subset of the interval [0, 1/2]. For example, let V be a Bernstein set: K ∩V ≠ ∅ and K \V ≠ ∅ for every uncountable closed set K ⊆ [0, 1/2]. Define S ⊆ IR^{2} by:

S = { (x, 0) : 0 ≤ x ≤ 1/2 } ∪ { (x, 1) : x ∈ V } ,

a subset of two horizontal intervals. Then in the metric space S, the diameter of a unit ball \overline{B}_{1}(x, 0) is either ≥ 1 or ≤ 1/2, according as x ∈ V or not. So diam \overline{B}_{1}(x, 0) is a non-measurable function of the point (x, 0).

(5.3) Example. This example has non-measurable upper density \overline{Δ}_{μ}^{s}. It is a bit more involved than the others. It is not clear to me whether there is such an example where S is (isometric to) a subset of Euclidean space IR^{n}. (Of course, since

Figure 1. (a) labels; (b) ratios.

the example below is a zero-dimensional separable metric space, it is homeomorphic to a subset of the line IR.)

We will use the notation from my text [5, §4.3]. Begin with an alphabet E = {0, 1, 2, 3}. These letters label the edges of a directed multigraph (Figure 1). A compact space T of infinite strings is thus defined: Let T be the set of all infinite strings σ from the alphabet E where all letters (with possibly a single exception) are 0’s and 1’s.

Next we define a metric ρ on T. The recursive rules are:

ρ(0σ, 0τ) = ρ(σ, τ)/4
ρ(1σ, 1τ) = ρ(σ, τ)/4
ρ(2σ, 2τ) = ρ(σ, τ)
ρ(3σ, 3τ) = ρ(σ, τ)
ρ(0σ, 1τ) = 1
ρ(0σ, 2τ) = 1 + ρ(0σ, τ)/4
ρ(0σ, 3τ) = 1
ρ(1σ, 2τ) = 1 + ρ(1σ, τ)/4
ρ(1σ, 3τ) = 1
ρ(2σ, 3τ) = 2.

Also, ρ(σ, τ) = ρ(τ, σ) and ρ(σ, σ) = 0. Verify by cases that ρ satisfies the triangle inequality. If α ∈ {0, 1}^{(∗)} is a finite string of 0’s and 1’s, then the cylinder set [α] has diameter 4^{−|α|}·2. Indeed, if σ, τ ∈ [α] do not involve 2 or 3, then ρ(σ, τ) ≤ 4^{−|α|}; except for (α2σ′, α3τ′) the maximum is 4^{−|α|} + ρ(σ′, τ′)/4 for certain σ′, τ′ ∈ [α], so it is ≤ 4^{−|α|} + 4^{−|α|} · 2/4 = 4^{−|α|} · (3/2).

Now T is a “graph self-similar” set. It is made up of four parts: two parts A = [2], B = [3], which are (1/4, 1/4)-Cantor sets; and two parts [0], [1] similar to all of T, shrunk by a factor 1/4.

Next define a measure μ by defining it on the basic cylinder sets [α]. If α, β ∈ {0, 1}^{(∗)} are finite strings of 0’s and 1’s, let:

μ[α] = 4^{−|α|}; μ[α2β] = 0; μ[α3β] = 4^{−|α|} · 2^{−1−|β|}.

Figure 2. A Markov chain.

The measure may be thought of as constructed by a Markov chain with probabilities shown in Figure 2. Note that μ(T \{0, 1, 3}^{(ω)}) = 0: the set of strings with a letter 2 has measure 0.

Now the subset K = {0, 1}^{(ω)} contained in T is a copy of the Cantor set (with metric 1/4, 1/4). Let V be a non-measurable subset of it, say a Bernstein set. Define a subset S ⊆ T by removing all strings α2σ′ where ασ′ ∉ V. Then S will be our (noncompact) metric space. We throw out only a subset of T \ {0, 1, 3}^{(ω)}, a closed set of measure 0, so μ is still a regular Borel measure on the remainder S.

Consider a ball \overline{B}_{r}(σ) in S, where σ ∈ K ⊆ S. Choose the integer k so that 4^{−k} ≤ r < 4^{−k+1}. Write α for the first k letters of σ, and σ′ for the rest of σ (so σ = ασ′). If 4^{−k} < r < 4^{−k+1}, then

\overline{B}_{r}(σ) ⊆ [α]
\overline{B}_{r}(σ) ⊇ [α0] ∪ [α1] ∪ [α3] ∪ [α2β]

for a long enough prefix β of σ′. If r = 4^{−k}, then

\overline{B}_{r}(σ) = [α0] ∪ [α1] ∪ [α3] ∪ \{α2σ′\}  if σ ∈ V
\overline{B}_{r}(σ) = [α0] ∪ [α1] ∪ [α3]  if σ ∉ V .

So, for 4^{−k} < r < 4^{−k+1},

μ(\overline{B}_{r}(σ)) = 4^{−k},  diam \overline{B}_{r}(σ) = 4^{−k} · 2,  \frac{μ(\overline{B}_{r}(σ))}{diam \overline{B}_{r}(σ)} = \frac{1}{2}.

And for r = 4^{−k}, μ(\overline{B}_{r}(σ)) = 4^{−k} and diam \overline{B}_{r}(σ) = 4^{−k} · 2 if σ ∈ V but diam \overline{B}_{r}(σ) ≤ 4^{−k} · (3/2) if σ ∉ V so that

\frac{μ(\overline{B}_{r}(σ))}{diam \overline{B}_{r}(σ)} = \frac{1}{2} if σ ∈ V  but  \frac{μ(\overline{B}_{r}(σ))}{diam \overline{B}_{r}(σ)} ≥ \frac{2}{3} if σ ∉ V .

Thus

\{ σ ∈ K : \overline{Δ}_{μ}^{1}(σ) < \frac{2}{3} \} = V

is a non-measurable set.

Note that 𝒯^{1}(S) = 0 since dim T = 1/2. But \overline{Δ}_{μ}^{1}(σ) = ∞ for σ that involve a letter 3, so this measure μ does not contradict (4.4 b) or (4.5 a).

6. Hausdorff Measure

The Hausdorff measures ℋ^{s} may be treated in a similar way. The derivation basis must be changed, however. The case for the metric space IR was treated by Thomson [14].

Let (S, ρ) be a separable metric space. A constituent is a pair (B, x), where B is a closed bounded set with at least two points [or a single isolated point of S], and x ∈ B. A packing is a disjoint collection π of consitutents. A closed cover of a set A is a collection β of constituents (B, x) with x ∈ A such that A ⊆ ⋃_{(B,x)∈β} B. A fine cover of A is a collection β of constituents such that, for every x ∈ A, and every ε > 0, there is B such that (B, x) ∈ β, x ∈ A, and diam B < ε.

The Banach argument in the Vitali theorem fits this setting [8, Theorem 1.10].

(6.1) Theorem. Let (S, ρ) be a separable metric space, let E ⊆ S be a set, let β be a fine cover of E, and let s > 0. Then there is a (finite or infinite) packing π = {(B_{i}, x_{i})} ⊆ β such that:

either  ∑_{i}(diam B_{i})^{s} = ∞  or  ℋ^{s} (E \ ⋃_{i} B_{i}) = 0.

Let s be a positive number, and let A ⊆ S be a set. For ε > 0, define

ℋ_{ε}^{s}(A) = inf ∑_{(B,x)∈β}(diam B)^{s},

where the infimum is over all countable closed covers β of A. The s-dimensional Hausdorff outer measure of a set A is

ℋ^{s}(A) = lim_{ε→0} ℋ_{ε}^{s}(A) = sup_{ε>0} ℋ_{ε}^{s}(A).

The closure of a set has the same diameter as the set itself. Thus, in the definition of Hausdorff measure, we may consider only covers by closed sets.

Let μ be a finite Borel measure. The upper s-density required for this basis, defined for non-isolated points x ∈ S, is:

\overline{d}_{μ}^{s}(x) = lim_{r→0} (sup \{ \frac{μ(B)}{(diam B)^{s}} : B closed, x ∈ B, 0 < diam B < r \}) .

If S is Euclidean space, then since the convex hull of a closed set is a closed convex set with the same diameter, the same upper density is obtained if we restrict the sets B to convex sets; thus \overline{d}_{μ}^{s} is sometimes known as the upper convex density [8, p. 21].

(6.2) Proposition. The upper density \overline{d}_{μ}^{s}(x) is a Borel measurable function of x.

Proof. First note that open sets may be used in the definition:

\overline{d}_{μ}^{s}(x) = lim_{r→0} (sup \{ \frac{μ(U)}{(diam U)^{s}} : U open, x ∈ U, 0 < diam U < r \}) .

Indeed, for any closed set B, the 1/n-neighborhood U_{n} of B decreases to B, and diam U_{n} → diam B; μ(U_{n}) → μ(B). And for any open set U, the set B_{n} of points with distance at least 1/n from the complement is closed, B_{n} increases to U, and diam B_{n} → diam U; μ(B_{n}) → μ(U).

Now for fixed r > 0, write

θ_{r}(x) = sup \{ \frac{μ(U)}{(diam U)^{s}} : U open, x ∈ U, 0 < diam U < r \} .

We claim that θ_{r} is a Borel function. In fact, for each t, the set L_{t} = { x : θ_{r}(x) > t } is an open set. To see this, suppose x_{0} ∈ L_{t}. Then there is an open set U with x_{0} ∈ U, 0 < diam U < r, and

\frac{μ(U)}{(diam U)^{s}} > t.

But then, for any x ∈ U, the same set U shows that θ_{r}(x) > t. Thus L_{t} is open.

Finally, \overline{d}_{μ}^{s}(x) = lim_{n} θ_{1/n}(x) is the pointwise limit of a sequence of Borel functions, so it is Borel. □

If h is a non-negative real-valued constituent function, the fine variation is defined much as before: if β is a closed cover, then

v_{β}(h) = sup ∑_{(B,x)∈π} h(B, x),

where the supremum is over all packings π ⊆ β; the fine variation of h is

v_{∗}(h) = inf v_{β}(h),

where the infimum is over all fine covers β. When h(B, x) = f(x)(diam B)^{s}, write v_{β}^{s}(f) = v_{β}(h) and v_{∗}^{s}(f) = v_{∗}(h).

The following results are proved in the same way as before. Replace 𝒯^{s} by ℋ^{s}, “centered cover” by “closed cover”, V_{β} by v_{β}, \overline{Δ}_{μ}^{s} by \overline{d}_{μ}^{s}, and so on. The Vitali theorem (6.1) should be used.

(6.3) Theorem. Let h be a nonnegative constituent function. Then μ defined by μ(A) = v_{∗}(h 1l_{A}) is a metric outer measure on S.

(6.4) Theorem. Let (S, ρ) be a separable metric space, and let s > 0. Then for every set E ⊆ S, we have v_{∗}^{s}(1l_{E}) = ℋ^{s}(E).

In part (c), define

F_{n} = \{ x ∈ E_{1} : \frac{μ(B)}{(diam B)^{s}} < α^{−2} for all B with x ∈ B, diam B < \frac{1}{n} \} .

(6.5) Proposition. Let S be a separable metric space, let s > 0, and let f be a nonnegative real-valued Borel function on S. Then

v_{∗}^{s}(f) = ∫_{S} f(x) ℋ^{s}(dx).

(6.6) Theorem. (a) Let μ be a finite Borel measure on the separable metric space S and let E ⊆ S be a Borel set. Then

μ(E) ≥ ∫_{E} \overline{d}_{μ}^{s}(x) ℋ^{s}(dx).

(b) If, in addition, ℋ^{s}(E) < ∞, and \overline{d}_{μ}^{s} < ∞ on E, then

μ(E) = ∫_{E} \overline{d}_{μ}^{s}(x) ℋ^{s}(dx).

The following corollaries are known; see [1], [15], [8, Theorems 2.3, 2.4].

(6.7) Corollary. Let S be a separable metric space, let s > 0, and let E ⊆ S be a Borel set with ℋ^{s}(E) < ∞.

(a) Let μ be a finite Borel measure. When the products are well-defined [i.e. not 0 · ∞], we have

ℋ^{s}(E) inf_{x∈E} \overline{d}_{μ}^{s}(x) ≤ μ(E) ≤ ℋ^{s}(E) sup_{x∈E} \overline{d}_{μ}^{s}(x).

(b) The density theorem: Let μ be the restriction of ℋ^{s} to E. Then \overline{d}_{μ}^{s}(x) = 1 for ℋ^{s}-almost every x ∈ E and \overline{d}_{μ}^{s}(x) = 0 for ℋ^{s}-almost every x ∉ E.

7. Open-ball diameter covering measure

Let us consider an alternative for the diameter covering measure. We are interested in using open balls in the centered covers and in the upper density. Since closed balls appear in the Vitali theorem (2.1), some care must be taken.

A simple variant of the lim sup and lim inf will be used. They are the limsup and liminf if we ignore countable sets.

Suppose q(r) ∈ IR is defined for each r > 0. Then

(a) u lim sup_{r→0} q(r) is the infimum of all α such that, for some η > 0, we have q(r) < α for all but countably many r with 0 < r < η.

(b) u lim sup_{r→0} q(r) ≥ α means: for all ε > 0 and all η > 0, there are uncountably many r with 0 < r < η and q(r) > α − ε.

(c) u lim inf_{r→0} q(r) is the supremum of all α such that, for some η > 0, we have q(r) > α for all but countably many r with 0 < r < η.

(d) u lim inf_{r→0} q(r) ≤ α means: for all ε > 0 and all η > 0, there are uncountably many r with 0 < r < η and q(r) < α + ε.

Let us begin with the density. First, note that the lower density is the same for open and closed sets.

(7.1) Proposition. Let (S, ρ) be a metric space, let x ∈ S be a non-isolated point, let μ be a finite Borel measure, and let s > 0. Then the following four values are all equal:

D_{1} = lim inf_{r→0} \frac{μ(B_{r}(x))}{( diam B_{r}(x))^{s}}

D_{2} = u lim inf_{r→0} \frac{μ(B_{r}(x))}{( diam B_{r}(x))^{s}}

D_{3} = lim inf_{r→0} \frac{μ(\overline{B}_{r}(x))}{( diam \overline{B}_{r}(x))^{s}}

D_{4} = u lim inf_{r→0} \frac{μ(\overline{B}_{r}(x))}{( diam \overline{B}_{r}(x))^{s}} .

Proof. The relevant facts are: B_{r+ε}(x) ↓ \overline{B}_{r}(x) as ε ↓ 0, so (i) μ(B_{r+ε}(x)) → μ(\overline{B}_{r}(x));

and \overline{B}_{r−ε}(x) ↑ B_{r}(x), so (ii) μ(\overline{B}_{r−ε}(x)) → μ(B_{r}(x)); (iii) diam \overline{B}_{r−ε}(x) → diam B_{r}(x). To begin, D_{1} ≤ D_{2} and D_{3} ≤ D_{4} are clear. Next, suppose α > D_{1}. For η > 0, there is r with 0 < r < η and

\frac{μ(B_{r}(x))}{( diam B_{r}(x))^{s}} < α.

Use (ii) and (iii): if ε > 0 is small enough, then

\frac{μ(\overline{B}_{r−ε}(x))}{( diam \overline{B}_{r−ε}(x))^{s}} < α.

Thus we have D_{4} ≤ α. This holds for any α > D_{1}, so this means D_{4} ≤ D_{1}. Next suppose α > D_{3}. For η > 0, there is r with 0 < r < η and

\frac{μ(B_{r}(x))}{( diam B_{r}(x))^{s}} < α.

If ε > 0 is small enough, then μ(B_{r+ε}(x)) ≈ μ(\overline{B}_{r}(x)) and diam B_{r+ε}(x) ≥ diam \overline{B}_{r}(x), so

\frac{μ(\overline{B}_{r+ε}(x))}{( diam \overline{B}_{r+ε}(x))^{s}} < α.

Thus we have D_{2} ≤ α. Therefore D_{2} ≤ D_{3}.

To summarize: D_{1} ≤ D_{2} ≤ D_{3} ≤ D_{4} ≤ D_{1}, so they are all equal. □

The lower density \underline{Δ}_{μ}^{s}(x) is defined by any of the four expressions. It is a Borel measurable function: for a fixed r, the functions x ↦ μ(B_{r}(x)) and x ↦ diam B_{r}(x) are both lower semicontinuous, hence measurable. For fixed x, the functions r ↦ μ(B_{r}(x)) and r ↦ diam B_{r}(x) are continuous from the left, so the lim inf called D_{1} above is the same when computed using only rational r. So \underline{Δ}_{μ}^{s} is a measurable function of x.

An example (5.2) shows that things are not as nice for the upper density.

(7.2) Proposition. Let (S, ρ) be a metric space, let x ∈ S be a non-isolated point, let μ be a finite Borel measure, and let s > 0. Write:

D_{1} = lim sup_{r→0} \frac{μ(B_{r}(x))}{( diam B_{r}(x))^{s}}

D_{2} = u lim sup_{r→0} \frac{μ(B_{r}(x))}{( diam B_{r}(x))^{s}}

D_{3} = lim sup_{r→0} \frac{μ(\overline{B}_{r}(x))}{( diam \overline{B}_{r}(x))^{s}}

D_{4} = u lim sup_{r→0} \frac{μ(\overline{B}_{r}(x))}{( diam \overline{B}_{r}(x))^{s}} .

Then D_{1} = D_{2} = D_{4} ≤ D_{3}.

Proof. First, D_{1} ≥ D_{2} and D_{3} ≥ D_{4} are clear.

Next, suppose α < D_{1}. For η > 0, there is r with 0 < r < η and

\frac{μ(B_{r}(x))}{( diam B_{r}(x))^{s}} > α.

For ε > 0 small enough,

\frac{μ(\overline{B}_{r−ε}(x))}{( diam \overline{B}_{r−ε}(x))^{s}} > α.

Thus D_{4} ≥ α. This proves D_{4} ≥ D_{1}.

Now as ε decreases to 0, the open ball B_{r+ε}(x) decreases to the closed ball \overline{B}_{r}(x). But diam B_{r+ε}(x) need not converge to diam \overline{B}_{r}(x). However, diam B_{r}(x) is a monotone function of r, so it is continuous at all but countably many values of r. Wherever it is continuous,

diam B_{r}(x) = diam \overline{B}_{r}(x) = lim_{ε→0} diam B_{r+ε}(x).

Next let α < D_{4}. For η > 0 there are uncountably many r with 0 < r < η and

\frac{μ(\overline{B}_{r}(x))}{( diam \overline{B}_{r}(x))^{s}} > α.

In particular, there is such an r where diam \overline{B}_{r}(x) = lim_{ε→0} diam B_{r+ε}(x). So for ε > 0 small enough,

\frac{μ(B_{r+ε}(x))}{( diam B_{r+ε}(x))^{s}} > α.

Thus, as usual, D_{2} ≥ D_{4}.

In summary, D_{2} ≥ D_{4} ≥ D_{1} ≥ D_{2} so these three are all equal. Also, D_{3} ≥ D_{4}, but this inequality may be strict. □

Write \overline{𝒟}_{μ}^{s}(x) for the upper density defined by D_{1} = D_{2} = D_{4}. Clearly \overline{𝒟}_{μ}^{s}(x) ≤ \overline{Δ}_{μ}^{s}(x). But (because it is defined by D_{1}) \overline{𝒟}_{μ}^{s} is a Borel measurable function of x.

The covering measure to use in this connection is defined with centered covers by open balls. This measure differs from 𝒞^{s} (and from ℋ^{s} and 𝒯^{s}) by at most a factor 2^{s}. In particular, measure 0 and finite measure are the same as for the other measures.

Now consider the Vitali theorem (6.1). A constituent will be a pair (B_{r}(x), x); an open ball and its center. Let us say that a collection β of open constituents is a very fine cover of a set A iff for every x ∈ A and every ε > 0, there exist uncountably many r with 0 < r < ε and (B_{r}(x), x) ∈ β. If β is a very fine cover of a set E, then

β′ = \{ (\overline{B}_{r}(x), x) : (B_{r}(x), x) ∈ β, diam B_{r}(x) = diam \overline{B}_{r}(x) \}

is a fine cover of E by closed balls. Thus, (2.1) may be applied to β′. We get {B_{1}, B_{2}, ⋯ } ⊆ β with either ∑(diam B_{i})^{s} = ∞ [open or closed balls the same] or 𝒯^{s}(E \ ⋃_{i} \overline{B}_{i}) = 0; here we get only closed balls. But the result may still be used in the applications (3.1) and (3.3), provided we use very fine covers. In a very fine cover, we may delete all balls B_{r}(x) with diam B_{r}(x) ≠ diam \overline{B}_{r}(x), and if μ is a finite Borel measure, we may similarly delete all balls B_{r}(x) with μ(B_{r}(x)) ≠ μ(\overline{B}_{r}(x)).

The very fine variation will be defined as

inf V_{β}(h)

with infimum over all very fine covers β (by centered open balls).

The results of §4 should remain correct, using the density \overline{𝒟}_{μ}^{s}, the very fine variation, and the covering measure defined with centered open ball covers. Because the upper density \overline{𝒟}_{μ}^{s} is Borel measurable, there is no need to use upper and lower integrals.

References

1. A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions, Proc. Cambridge Philos. Soc. 41 (1945), 103–110.

2. A. S. Besicovitch and P. A. P. Moran, The measure of product and cylinder sets, J. London Math. Soc. 20 (1945), 110–120.

3. C. Carathéodory, Über das lineare maß von Punktmengen—eine Verallgemeinerung des Längenbegriffs, Nachrichten der K. Gesellshaft der Wissenschaften zu Göttingen, Mathematischphysikalische Klasse 4 (1914), 494–426, English translation in [6], pp. 47–73.

4. D. L. Cohn, Measure Theory, Birkhäuser, Boston, 1980.

5. G. A. Edgar, Measure, Topology, and Fractal Geometry, Springer-Verlag, New York, 1990.

6. G. A. Edgar, Classics on Fractals, Addison-Wesley, Reading, Mass., 1993.

7. G. A. Edgar, Packing measure as a gauge variation, Proc. Amer. Math. Soc. 122 (1994), 167–174.

8. K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1985.

9. K. Kuratowski, Topology, vol. I, Academic Press, New York, 1966.

10. S. Meinershagen, The symmetric derivation basis measure and the packing measure, Proc. Amer. Math. Soc. 103 (1988), 813–814.

11. C. A. Rogers, Hausdorff Measures, Cambridge University Press, Cambridge, 1970.

12. X. Saint Raymond and C. Tricot, Packing regularity of sets in n-space, Math. Proc. Cambridge Phlos. Soc. 103 (1988), 133–145.

13. B. S. Thomson, Construction of measures in metric spaces, J. London Math. Soc. 14 (1976), 21–24.

14. B. S. Thomson, Derivates of interval functions, Memoirs of the Amer. Math. Soc. 452, American Mathematical Society, Providence, 1991.

15. H. Wallin, Hausdorff measures and generalized differentiation, Math. Ann. 183 (1969), 275– 286.

Department of Mathematics, The Ohio State University, Columbus, Ohio 43210 E-mail address: edgar@math.ohio-state.edu