PACKING MEASURE AS A GAUGE VARIATION

G. A. Edgar

The Ohio State University

revised April, 1993

Abstract. Meinershagen noted that (in the line) the fractal packing measure of Tricot and Taylor can be considered to be a Henstock-Thomson gauge variation (“method III”) for an appropriate choice of derivation basis and set function. We show that this point of view remains interesting in a general metric space.

0. Introduction

The best-known fractal measure is the Hausdorff measure. Tricot and Taylor [13, 10] propose the use of an additional fractal measure known as the packing measure. Fractals should be sets with some weak regularity properties, such as self-similarity. Taylor proposes that the term “fractal” should mean a set where the Hausdorff and packing measures both yield the same fractal dimension. A good reference for the packing measure is [9]. For packing measure in general metric spaces, see [2], [4].

Henstock [6] defined a family of “variation” set functions. Thomson [11] clarified the definition, calling it “method III” (following Munroe’s methods I and II). In [12], he notes that if the variation is computed starting from the set function (diam B)^{s}, where s > 0 is a given positive number, then the resulting variation is an outer measure with the properties appropriate for an s-dimensional fractal measure. Thomson asked whether (in the line IR) this variation is the Hausdorff outer measure. Meinershagen [8] answered that this measure (the “symmetric derivation basis measure” computed with the set function (diam B)^{s}) is the packing measure, not the Hausdorff measure. Meinershagen did her computations on the line IR. In this paper we show that in other metric spaces as well, the packing measure is a Henstock-Thomson gauge variation.

I will discuss “covering measure” and “Hausdorff measure” from this point of view in a subsequent paper.

Typeset by AMS-TEX

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

Key words and phrases. fractal measure, packing measure, gauge integral, Henstock variation, derivation basis.

1. Packing measure

1A. Tricot’s Definition. We begin with the definition of the packing measure (see [13, 10, 9]). Let (S, ρ) be a metric space. For x ∈ S and r > 0, the open ball is

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 packing of A is a countable disjoint collection of open balls with center in A:

{B_{r_{1}}(x_{1}), B_{r_{2}}(x_{2}), ⋯ } ,

where x_{i} ∈ A and B_{r_{i}}(x_{i}) ∩ B_{r_{j}}(x_{j}) = ∅ for i ≠ j.

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

𝒫_{ε}^{s}(A) = sup ∑_{i} ( diam B_{r_{i}}(x_{i}))^{s},

where the supremum is over all packings of A by centered balls with diameter ≤ ε. [Of course, in Euclidean space diam B_{r}(x) = 2r, but in a general metric space this need not be true.] The s-dimensional packing pre-measure of A is

𝒫̃^{s}(A) = lim_{ε→0} 𝒫_{ε}^{s}(A).

The s-dimensional packing outer measure is the outer measure 𝒫^{s} defined from the set function 𝒫̃^{s} by method I. That is,

𝒫^{s}(A) = inf { ∑_{D∈𝒟} 𝒫̃^{s}(D) : 𝒟 is a countable cover of A } .

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

The definition is a bit awkward to use, because of the “method I” step added on the end. But the set function 𝒫̃^{s} is not countably subadditive, so this final step is needed to obtain a genuine outer measure.

Here are a few observations about the definition: (1.2) Finite packings suffice. That is,

𝒫_{ε}^{s}(A) = sup ∑_{i} ( diam B_{r_{i}}(x_{i}))^{s},

where the supremum is over all finite packings of A by centered balls with diameter ≤ ε.

(1.3) An open ball B_{r}(x) is the increasing union of a sequence \overline{B}_{r−1/n}(x) of closed balls, and diam \overline{B}_{r−1/n}(x) → diam B_{r}(x). In many metric spaces, diam B_{r}(x) = diam \overline{B}_{r}(x). If that is true, then the same value of 𝒫̃^{s} is obtained if we use packings by closed balls instead of open balls. From a remark of Haase [4], it seems that this equality is false in general.

(1.4) If A ⊆ S is a set, and \overline{A} is the closure of the set, then 𝒫̃^{s}(A) = 𝒫̃^{s}( \overline{A}). Indeed, on the one hand, any centered-ball packing of A is also a centeredball packing of \overline{A}, so 𝒫̃^{s}(A) ≤ 𝒫̃^{s}( \overline{A}). On the other hand, if B_{r}(x) is a ball with x ∈ \overline{A}, then there is a point x′ ∈ A as close as we like to x and a radius r′ = r − ρ(x′, x) so that B_{r′}(x′) ⊆ B_{r}(x) and diam B_{r′}(x′) is as close as we like to diam B_{r}(x). Thus, given a centered-ball packing \{B_{r_{i}}(x_{i})\} of \overline{A}, we may choose a centered-ball packing {B_{r′_{i}}(x′_{i})} of A with ∑(diam B_{r′_{i}}(x′_{i}))^{s} as close as we like to ∑(diam B_{r_{i}}(x_{i}))^{s}. So 𝒫̃^{s}(A) = 𝒫̃^{s}( \overline{A}).

(1.5) Since 𝒫̃^{s}(A) = 𝒫̃^{s}( \overline{A}), in the definition

𝒫^{s}(A) = inf { ∑_{D∈𝒟} 𝒫̃^{s}(D) : 𝒟 is a countable cover of A } .

of 𝒫^{s}, it is enough to use covers 𝒟 consisting of closed sets. Because of this fact, for any set E ⊆ S, there is a Borel set A (in fact an F_{σδ}) with A ⊇ E and 𝒫^{s}(A) = 𝒫^{s}(E).

(1.6) If E_{n} be an increasing sequence of (possibly non-measurable) sets, then we have lim_{n} 𝒫^{s}(E_{n}) = 𝒫^{s}( ⋃ E_{n}). Indeed, for each E_{n} choose a Borel set A_{n} ⊇ E_{n} with the same outer measure, then consider ⋃_{n} A_{n}.

(1.7) Given a set A, there is a critical value s_{0} ∈ [0, ∞] such that

𝒫^{s}(A) = \{ \begin{matrix}∞, & for all s with s < s_{0}, \\ 0, & for all s with s > s_{0}. \end{matrix}

This critical value is the packing dimension of the set A, which will be written Dim A = s_{0}. At least in Euclidean space, the packing dimension coincides with the metrische Dimensionszahl of Wegmann [14, §3.3].

1B. Henstock’s variation. The variation can be defined for a general “derivation basis” (Henstock [6]). Thomson [11] calls the construction “method III”. Here we will use only the “centered ball” basis, so reference to the basis will be supressed from our notation and terminology.

Let (S, ρ) be a metric space. A constituent is a pair (B, x), where x ∈ S is a point, and B is an open ball B_{r}(x) with center x and positive radius r. In many metric spaces, the center x is uniquely determined by the ball B, but this fails in some metric spaces (such as ultrametric spaces), so the intended center is included as the second component of the constituent.

Let E ⊆ S. A packing of E is a countable set

π = {(B_{1}, x_{1}), (B_{2}, x_{2}), ⋯ }

of constituents with x_{i} ∈ E for all i and B_{i} ∩ B_{j} = ∅ for i ≠ j. The term packing by itself means a packing of the whole space S. A gauge on E is a function Δ: E → (0, ∞). If Δ is a gauge for E and π is a packing of E, then we say π is Δ-fine iff diam B ≤ Δ(x) for all constituents (B, x) of π.

(1.8) Definition. Let s be a positive number. For a gauge Δ, define

𝒫_{Δ}^{s}(A) = sup ∑_{(B,x)∈π} ( diam B)^{s},

where the supremum is over all Δ-fine packings π of A. As Δ decreases toward 0, the value 𝒫_{Δ}^{s}(A) decreases. For the limit, write:

𝒫_{•}^{s}(A) = inf_{Δ} 𝒫_{Δ}^{s}(A).

The positive constants ε are among the gauges, and if Δ(x) = ε for all x, the notation 𝒫_{Δ}^{s} is consistent with the previous notation 𝒫_{ε}^{s}.

This time we get an outer measure without an extra “method I” step at the end. This is a special case of the general theory of [11], but we will prove it here because of its central importance.

(1.9) Theorem. The set function 𝒫_{•}^{s} is a metric outer measure.

Proof. The only packing of the empty set is the empty packing, so 𝒫_{Δ}^{s}(∅) = 0 for all Δ, and 𝒫_{•}^{s}(∅) = 0.

If E ⊆ F, any packing of E is also a packing of F, so 𝒫_{Δ}^{s}(E) ≤ 𝒫_{Δ}^{s}(F) for all Δ, and thus 𝒫_{•}^{s}(E) ≤ 𝒫_{•}^{s}(F).

Suppose E = ⋃_{n=1}^{∞} E_{n}. I must show that 𝒫_{•}^{s}(E) ≤ ∑_{n} 𝒫_{•}^{s}(E_{n}). If the righthand-side is ∞, there is nothing to prove, so suppose it is finite. Fix ε > 0. For each n, choose a gauge Δ_{n} on E_{n} so that 𝒫_{Δ_{n}}^{s}(E_{n}) ≤ 𝒫_{•}^{s}(E_{n}) + ε/2^{n}. Define Δ by: Δ(x) = Δ_{n}(x) if x ∈ E_{n} \ ⋃_{i=1}^{n−1} E_{i}. Then Δ is a gauge on E. Let π be a Δ-fine packing of E. Then for each n,

π_{n} = { (B, x) ∈ π : x ∈ E_{n} \ ⋃_{i=1}^{n−1} E_{i} }

is a Δ_{n}-fine packing of E_{n}. So

∑_{(B,x)∈π}(diam B)^{s} = ∑_{n} ∑_{(B,x)∈π_{n}}(diam B)^{s}

≤ ∑_{n} 𝒫_{Δ_{n}}^{s}(E_{n}) ≤ ε + ∑_{n} 𝒫_{•}^{s}(E_{n}).

Therefore 𝒫_{•}^{s}(E) ≤ 𝒫_{Δ}^{s}(E) ≤ ε + ∑ 𝒫_{•}^{s}(E_{n}). Let ε → 0 to obtain 𝒫_{•}^{s}(E) ≤ ∑ 𝒫_{•}^{s}(E_{n}).

Suppose E, F ⊆ S have positive separation, dist(E, F) = ε > 0. Let Δ be any gauge with Δ(x) ≤ ε/2. Then the constituents of any Δ-fine packing of E are disjoint from the constituents of any Δ-fine packing of F. So the union of the two packings is a Δ-fine packing of E ∪ F. Therefore 𝒫_{Δ}^{s}(E ∪ F) = 𝒫_{Δ}^{s}(E) + 𝒫_{Δ}^{s}(F). This is true for all sufficiently small gauges, so 𝒫_{•}^{s}(E ∪ F) = 𝒫_{•}^{s}(E) + 𝒫_{•}^{s}(F). □

Now in fact, the variation outer measure 𝒫_{•}^{s} is just the packing outer measure 𝒫^{s}; the notation 𝒫_{•}^{s} was introduced for temporary use.

(1.10) Theorem. For any E ⊆ S and s > 0, we have 𝒫_{•}^{s}(E) = 𝒫^{s}(E).

Proof. The positive constants ε are among the gauges, so

𝒫̃^{s}(E) = inf_{ε} 𝒫_{ε}^{s}(E) ≥ inf_{Δ} 𝒫_{Δ}^{s}(E) = 𝒫_{•}(E).

If E ⊆ ⋃_{n} E_{n}, then 𝒫_{•}^{s}(E) ≤ 𝒫_{•}^{s}(⋃ E_{n}) ≤ ∑_{n} 𝒫_{•}^{s}(E_{n}) ≤ ∑_{n} 𝒫̃^{s}(E_{n}). Take the infimum over all such covers to conclude that 𝒫_{•}^{s}(E) ≤ 𝒫^{s}(E).

On the other hand, suppose Δ is a gauge on a set E. For each positive integer n let

E_{n} = \{ x ∈ E : Δ(x) ≥ \frac{1}{n} \} .

The sets E_{n} increase to E. Now for each n,

𝒫_{Δ}^{s}(E) ≥ 𝒫_{Δ}^{s}(E_{n}) ≥ 𝒫_{1/n}^{s}(E_{n}) ≥ 𝒫̃^{s}(E_{n}) ≥ 𝒫^{s}(E_{n}).

So by (1.6) we have 𝒫_{Δ}^{s}(E) ≥ 𝒫^{s}(E). This is true for all gauges Δ, so 𝒫_{•}^{s}(E) ≥ 𝒫^{s}(E). □

2. Gauge integral

If f is a nonnegative Borel function, then the integral of f may be computed as a variation.

(2.1) Definition. Let h be a nonnegative real-valued constituent function: that is, for each constituent (B, x), let h(B, x) ∈ [0, ∞). If Δ is a gauge, write

V_{Δ}(h) = sup ∑_{(B,x)∈π} h(B, x),

where the supremum is over all Δ-fine packings π of S. (When Δ decreases, V_{Δ}(h) decreases.) The variation of h is

V (h) = inf V_{Δ}(h),

where the infimum is over all gauges Δ. When the constituent function h is of the special form

h(B, x) = f(x)(diam B)^{s},

for some nonnegative point-function f: S → [0, ∞), we will write V_{Δ}^{s}(f) = V_{Δ}(h) and V^{s}(f) = V (h). We will call V^{s}(f) the s-variation of f.

The following is proved in the same way as Theorem (1.9), above. Thomson calls μ a “method III” outer measure. We write 1l_{A} for the indicator function of a set A, and h 1l_{A} for the constituent function

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

(2.2) Theorem. Let h be a nonnegative real-valued constituent function. Then μ defined by μ(A) = V (h 1l_{A}) for all A ⊆ S is a metric outer measure on S.

For special h of the form f(x)(diam B)^{s}, the variation is the integral with respect to the packing measure 𝒫^{s}.

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

V^{s}(f) = ∫_{S} f(x) 𝒫^{s}(dx).

Proof. First, by the respective definitions, if A ⊆ S we have V_{Δ}^{s}(1l_{A}) = 𝒫_{Δ}^{s}(A) and V^{s}(1l_{A}) = 𝒫^{s}(A). Clearly, if a ≥ 0 is a constant and f ≥ 0 is a function, then V^{s}(af) = aV^{s}(f).

Now both 𝒫^{s} 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) 𝒫^{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) 𝒫^{s}(dx) = ∫_{S} f(x) 𝒫^{s}(dx),

as required. □

Let A ⊆ S be a set. A centered Vitali cover of A is a set β of constituents such that for every x ∈ A and every ε > 0, there is (B, x) ∈ β with diam B ≤ ε. A Borel measure μ has the strong Vitali property iff for any Borel set A with μ(A) < ∞, and any centered Vitali cover β of A, there is a packing π ⊆ β with

μ (A \ ⋃_{(B,x)∈π} B) = 0.

In many metric spaces, every Borel measure has the strong Vitali property. Besicovitch [1] proved this for Euclidean space (see also [5, p. 114]).

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

\underline{D}_{μ}^{s}(x) = lim inf_{r→0} \frac{μ(B_{r}(x))}{( diam B_{r}(x))^{s}} .

In this definition, the same lower density is obtained if we use closed balls rather than open balls; and the lim inf is unchanged if we remove any countable set of r > 0. Saint Raymond and Tricot [9] prove that, in Euclidean space, if 𝒫^{s}(E) < ∞, then

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

Here is a strengthened form of the result.

(2.4) Theorem. Let μ be a finite Borel measure on the metric space S. For any Borel set E ⊆ S, we have

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

Equality holds if μ has the strong Vitali property, 𝒫^{s}(E) < ∞, and \underline{D}_{μ}^{s}(x) < ∞ on E.

Proof. Let E be a Borel set. Let U ⊇ E be an open neighborhood of it, and let c < 1 be a positive constant. For each x ∈ E, there is Δ(x) > 0 so that

Δ(x) < dist(x, S \ U)

and

\frac{μ(B_{r}(x))}{( diam B_{r}(x))^{s}} ≥ c\underline{D}_{μ}^{s}(x) for all r < Δ(x).

So Δ is a gauge on E. If π is a Δ-fine packing of E, then

∑_{(B,x)∈π} \underline{D}_{μ}^{s}(x) (diam B)^{s} ≤ \frac{1}{c} ∑_{(B,x)∈π} μ(B) ≤ \frac{1}{c}μ(U).

Therefore V^{s}(\underline{D}_{μ}^{s} 1l_{E}) ≤ V_{Δ}^{s}(\underline{D}_{μ}^{s} 1l_{E}) ≤ (1/c)μ(U). Take the infimum over all open U ⊇ E and all c < 1 to obtain V^{s}(\underline{D}_{μ}^{s} 1l_{E}) ≤ μ(E). But \underline{D}_{μ}^{s} is a Borel function, so this means

∫_{E} \underline{D}_{μ}^{s}(x) 𝒫^{s}(dx) ≤ μ(E).

Now suppose μ has the strong Vitali property and let E be a Borel set with 𝒫^{s}(E) < ∞. Let Δ be a gauge on E such that 𝒫_{Δ}^{s}(E) < ∞ and let ε > 0. Then

\{ (B, x) constituent : x ∈ E, diam B ≤ Δ(x), \frac{μ(B)}{(diam B)^{s}} ≤ \underline{D}_{μ}^{s}(x) + ε \}

is a centered Vitali cover of E. So we may choose from it a packing π of E with

μ (E \ ⋃_{(B,x)∈π} B) = 0.

Then we have

μ(E) ≤ ∑_{(B,x)∈π} μ(B) ≤ ∑_{(B,x)∈π} (\underline{D}_{μ}^{s}(x) + ε) (diam B)^{s} ≤ V_{Δ}^{s}(\underline{D}_{μ}^{s} 1l_{E}) + ε𝒫_{Δ}^{s}(E).

Take the infimum over Δ and ε to obtain μ(E) ≤ V^{s}(\underline{D}_{μ}^{s} 1l_{E}). □

Consider the example where S = [0, 1], μ is Lebesgue measure on [0, 1], and 0 < s < 1. Then \underline{D}_{μ}^{s}(x) = 0 for all x, but μ(S) > 0. This shows that the hypothesis 𝒫^{s}(E) < ∞ cannot be dropped in the theorem.

4. Questions

The point of view exploited here suggests other questions. What fractal measures will we get if we compute the Henstock gauge variation,

using these same constituent functions, but other common differentiation bases? For example: (4.1) Pairs (B, x), where B is an open ball and x ∈ B (but possibly not the

center). (4.2) Pairs (B, x), where B is an open ball, and x ∈ S is a point, possibly not in

B at all; “Δ-fine” means that B ⊆ B_{Δ(x)}(x) (McShane basis). (4.3) In Euclidean space, pairs (B, x) where B is a cube with sides parallel to the

axes. Hundreds of other bases can be found in [5].

C. Cutler [2] argues that the “radius packing” measure is superior to the “diameter packing” measure that has been used here. (4.4) Carry out this study for the “radius packing” measure. This should be done

by replacing the set-function B_{r}(x) ↦ (diam B_{r}(x))^{s} with the set-function B_{r}(x) ↦ (2r)^{s}.

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. C. Cutler, The density theorem and Hausdorff inequality for packing measure in general metric spaces (to appear).

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

4. H. Haase, The packing theorem and packing measure, Math. Nachr. 146 (1990), 77–84.

5. C. A. Hayes and C. Y. Pauc, Derivation and Martingales, Springer-Verlag, New York, 1970.

6. R. Henstock, Generalized integrals of vector-valued functions, Proc. London Math. Soc. 19 (1969), 509–536.

7. R. Henstock, The General Theory of Integration, Clarendon Press, Oxford, 1991.

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

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

10. S. J. Taylor and C. Tricot, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679–699.

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

12. B. S. Thomson, Derivation bases on the real line (I), Real Anal. Exchange 8 (1982), 67–207.

13. C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74.

14. H. Wegmann, Die Hausdorff-Dimension von kartesischen Produkten metrischer Räume, J. Reine Angew. Math. 246 (1971), 46–75.

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