Eötvös Loránd University of Sciences

Faculty of Sciences Department of Analysis

On the dimensions in elementary geometric measure theory

Thesis

Written by:  Supervisor:
Mikolt Jelencsics  Prof. György Petruska

..................................................................................................................................

May 1996

Abstract

This paper provides an account of the mathematics of real sets of fractional and integral Hausdorff dimension. It is primarily concerned with the geometric theory rather than with applications, that is, we do not investigate measures for the sake of the integral, but for their own right. One of the basic theorems says that an H^{m}-measurable set (m is integer) splits into a regular (or m-rectifiable) and an irregular (or m-unrectifiable) part, and we find that these two types of set exhibit markedly different properties. One of our aim is to characterize these sets, and thus to relate the measure theoretic and the descriptive topological ideas.

Contents

1 Introduction 2

2 Preliminaries 3 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Covering results . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 A preparatory theorem . . . . . . . . . . . . . . . . . . . . . . 6

3 Density theorems for Hausdorff measures 7 3.1 Two types of densities . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Elementary density bounds . . . . . . . . . . . . . . . . . . . . 8

4 The structure of integral-dimensional sets 12 4.1 m-rectifiability and m-unrectifiability . . . . . . . . . . . . . . 12 4.2 Rectifiability and measures in cones . . . . . . . . . . . . . . . 14 4.3 Linear approximation properties . . . . . . . . . . . . . . . . . 17 4.4 A lemma on projections of unrectifiable sets . . . . . . . . . . 19 4.5 Rectifiability, densities, and projections . . . . . . . . . . . . . 21 4.6 Two equivalence theorems on rectifiability . . . . . . . . . . . 27 4.7 Density and the characterization of regular 1-sets . . . . . . . 28

5 Structure of sets of non-integral dimension 34 5.1 s-sets with 0 < s < 1 . . . . . . . . . . . . . . . . . . . . . . . 34 5.2 Remarks on sets in higher dimensions . . . . . . . . . . . . . . 36

References 37

1 Introduction

We work in n-dimensional Euclidean space, IR^{n}, though many of the ideas apply equally to more general metric spaces. We also restrict our attention to Hausdorff measures corresponding to a numerical dimension s, rather than to an arbitrary function.

Section 2 only covers some preparatory lemmas and theorems.

In Section 3 we investigate the basic density properties. Such results were first given for linearly measurable sets by Besicovitch [1], [2]. There is no difficulty in generalizing this work to s-sets in IR^{n} for any s and n, so we do so.

In section 4 we deal with the rectifiability and unrectifiability of sets. The basic statement of Theorem 4.11 is that an m-rectifiable set has density 1 almost everywhere. Essentially the proof below with m = 2, n = 3 was given by Marstrand [6] and for general m and n by Mattila [8] . Kirchheim [5] showed that the statement (i) of Theorem 4.11 holds also for m-rectifiable sets in general metric spaces. His proof involves an interesting idea of using semi-norms induced by Lipschitz maps. For a more standard proof of (i) in IR^{n} see Federer [4] 3.2.19.

We present in detail the theory of linearly measurable sets in IR^{2}. This work is almost entirely due to Besicovitch [1], [2], the latter paper including some improved proofs as well as further results. Most of his proofs seem hard to better except in relatively minor ways and, hopefully, in presentation.

One of the most elegant theorems, Theorem 4.12 is stated at the last part of Section 4. In this theorem we compare regularity of an integral dimensional set with its rectifiability. Chlebík [3] improved this proposition by proving that there is a constant c(m) < 1, depending only on m and not on n, such that D^{m}(A, x) ≥ c(m) for H^{m}-almost all x ∈ A implies the m-rectifiability of A. Chlebík’s proof, which is based on the proofs in Marstrand [6] and Mattila [8], works also for subsets of infinite dimensional Hilbert spaces.

For general m very little is known about the best possible constant c(m). However, for m = 1 there is much more information. For subsets of IR^{2} Besicovitch showed that

D^{1}(A, x) ≥ 3 4 for H^{1}-almost all x ∈ A  (1)

implies the 1-rectifiability of A, see [2]. Preiss and Tišer [12] improved this by showing that 3/4 can be replaced by (2+√46)/12, however, their proof seems not to be correct. Besicovitch [1] gave an example of a 1-unrectifiable subset of IR^{2} for which D^{1}(A, x) = 1 2 for H^{1}-almost all x ∈ A and he conjectured that 3/4 could be replaced by 1/2 in (1). This conjecture is still open.

Most of my paper originates in Mattila [9], and later on I found Mattila [10] being very helpful, especially with the proofs.

A number of the proofs have been somewhat simplified from their original form. Further shortenings would undoubtedly be possible, but the author’s desire for perfection has had to be offset by the requirement to finish the paper in a finite time.

It is a great pleasure to record my gratitude to all those who have helped with this paper in any way. I am particularly indebted to András Lukács for his continuous encouragement. I should like to thank Zoltán Hidvégi and the Department of Computer Science for their technical support.

2 Preliminaries

2.1 Notation

We will generally follow the customary practice in the most standardized notation and terminology with one exception that should be kept in mind: following Federer we call measure what is usually called outer measure.

We first establish the basic notation used throughout this paper. The closed and open balls with centre x ∈ IR^{n} and radius r, 0 < r < ∞, are denoted by

B(x, r) = {y ∈ IR^{n} : |x − y| ≤ r}, U(x, r) = {y ∈ IR^{n} : |x − y| < r}

The diameter of a non-empty subset A of IR^{n} is

diam(A) = sup{|x − y| : x, y ∈ A}.

We agree on diam(∅) = 0.

λ^{n} denotes the n-dimensional Lebesgue (outer) measure, while α^{n} denotes the n-dimensional Lebesgue measure of the n-dimensional unit ball. We do not use its precise value, but do need it for brevity of notation. H^{s} and H_{δ}^{s} denote the s-dimensional non-normalized Hausdorff measure over IR^{n}, and its δ-premeasure, respectively.

For any set A we denote

A(δ) := {x ∈ IR^{n} : d(x, A) ≤ δ},

where d is the distance induced by the metric of IR^{n}.

A(n, m) denotes the set of m-dimensional affine subspaces of IR^{n}, whereas G(n, m) is the set of m-dimensional linear subspaces of IR^{n}.

A moment of reflection shows that the Hausdorff measurable sets are Lebesgue measurable and also the other way round, a Lebesgue measurable set is also Hausdorff measurable, thus sometimes we use the two expressions interchangeably.

An H^{s}-measurable set E ⊂ IR^{n} for which 0 < H^{s}(E) < ∞ is termed an s − set; a 1-set is sometimes called a linearly measurable set. Clearly, the Hausdorff dimension of an s-set equals s, but it is important to realize that an s-set is something much more specific than a measurable set of Hausdorff dimension s.

In general we exclude the case s = 0 as this often requires separate treatment from other values of s. However, since 0-sets are simply finite sets of points their properties are straightforward. On the other hand, most of the concepts and the propositions apply equally for s-dimensional sets of measure zero, therefore we do not restrict ourselves to sets of positive measure. Similarly, many of the proposition could be generalized with ease to ‘locally s-sets’, that is, those sets with σ-finite measure whose intersection with any bounded set is of finite measure.

2.2 Covering results

One of the most useful tools of geometric measure theory is the Vitali covering theorem. Given a ‘sufficiently large’ collection of sets that cover some set A, the Vitali theorem selects a disjoint subcollection that covers almost all of A.

We include the following lemma at this point because it illustrates the basic principle embodied in the proof of Vitali’s result, but in a simplified setting, and because we have frequent recourse to it.

Lemma 2.1. Let {B(x, r(x)), x ∈ A} be a family of closed balls contained in a bounded subset of IR^{n}. Then we may find a finite or countably infinite set {x_{1}, x_{2}, …} ⊆ A such that the balls B(x_{i}, r_{i}), r_{i} = r(x_{i}), are disjoint and

⋃_{x∈A} B(x, r(x)) ⊆ ⋃_{i=1}^{∞} B(x_{i}, 4r_{i}).

Proof. We select the {x_{i}} inductively. Let d_{0} := sup{r(x) : x ∈ A} and choose x_{1} from A with r(x_{1}) ≥ 2 3d_{0}. If x_{1}, …, x_{m} have been chosen let d_{m} := sup{r(x) : x ∈ A, B(x, r(x)) disjoint from ⋃_{1}^{m} B(x_{i}, r_{i})}. If d_{m} = 0 the process terminates. Otherwise choose x_{m+1} from A with r(x_{m+1}) ≥ 2 3d_{m} such that B(x_{m+1}, r(x_{m+1})) is disjoint from ⋃_{1}^{m} B(x_{i}, r_{i}). Certainly, these balls are disjoint; we claim that they also have the required covering property. If x ∈ A, then either x = x_{i} for some i, or B(x, r(x)) intersects some B(x_{i}, r_{i}) with 3 2r_{i} ≥ r(x). If this was not the case x would have been selected in preference to the first point x_{m} for which 3 2r_{m} < r(x). (Note that, by summing volumes, ∑ r_{i}^{2} < ∞ so that r_{i} → 0 as i → ∞ if infinitely many balls are selected.) In either case, B(x, r(x)) ⊂ B(x_{i}, 4r_{i}), giving the lemma.

Remarks. A trivial check establishes that the constant 4 in Lemma 2.1 might be replaced with 3 + ε where ε is an arbitrary positive number (take 1 − ε ; 2+ε instead of the 2 3 in the proof), and also that the conditions on the set of balls could have been lessened (e.g., we can have several balls with the same centre, or the centres might lie in an unbounded set). However, the lemma will suffice us in this weak form.

We are going to use the Vitali covering theorem in its following form:

Theorem 2.2. Let μ be a Radon measure on IR^{n}, A ⊂ IR^{n}, and B a family of closed balls such that every point of A is the centre of arbitrarily small balls of B, that is,

inf{r : B(x, r) ∈ B} = 0 for x ∈ A.

Then we may select a (finite or countable) disjoint sequence B_{i} from B such that

μ(A \ ⋃_{i} B_{i}) = 0.

Further, given ε > 0, we may also require that

∑_{i} μ(B_{i}) ≤ μ(A) + ε.

2.3 A preparatory theorem

We state now a theorem (a simple version of Sard’s theorem for Lipschitz maps) for later use. A very general form of the theorem can be found in Federer [4] 3.4.3.

Theorem 2.3. If f : IR^{m} → IR^{n} is a Lipschitz map, then

H^{m}({f(x) : dim(f′(x)IR^{m}) < m}) = 0.

(Here dim(f′(x)IR^{m}) is the topological (in this case also the Hausdorff) dimension of the linear subspace f′(x)IR^{m} of IR^{n}.)

Proof. Let 0 < R < ∞ and set

A := {x ∈ B(0, R) ⊂ IR^{m} : dim(f′(x)IR^{m}) < m}.

Let ε > 0 and L := Lip(f). If x ∈ A and W_{x} := f′(x)IR^{m} + f(x) = {f′(x)y + f(x) : y ∈ IR^{m}}, then for sufficiently small r > 0,

fB(x, r) ⊆ B(f(x), Lr) ∩ W_{x}(εr).

Since dim W_{x} ≤ m − 1 we have (with a constant c depending only on m),

H_{∞}^{m}(fB(x, r)) ≤ cεr(Lr)^{m−1}.

By the Vitali covering theorem we can find disjoint balls B_{i} := B(x_{i}, r_{i}) such that

λ^{m}(A \ ⋃_{i} B_{i}) = 0 and ∑_{i} λ^{m}(B_{i}) ≤ λ^{m}(A) + ε.

Then fA ⊆ (⋃_{i} fB_{i}) ∪ f(A \ ⋃_{i} B_{i}) and H^{m}(f(A \ ⋃_{i} B_{i})) = 0. Hence

H_{∞}^{m}(fA) ≤ ∑_{i} H_{∞}^{m}(fB_{i}) ≤ cεL^{m−1} ∑_{i} r_{i}^{m} ≤ L^{m−1} cε ; α^{m} ∑_{i} λ^{m}(B_{i})
≤ L^{m−1} cε ; α^{m} (λ^{m}(A) + ε).

The observation that ε and R were arbitrary and λ^{m}(A) is finite completes the proof.

3 Density theorems for Hausdorff measures

3.1 Two types of densities

First we define the basic set densities that play a major rôle in our development. These densities are natural analogues of the Lebesgue-density, though their behaviour can be very different.

Definition 3.1. Let A ⊂ IR^{n}, x ∈ IR^{n}, we write

D^{s}(A, x) := lim sup_{r→0+} H^{s}(A ∩ B(x, r)) ; (2r)^{s},

D^{s}(A, x) := lim inf_{r→0+} H^{s}(A ∩ B(x, r)) ; (2r)^{s};

D^{s}(A, x), D^{s}(A, x) will be called the upper and lower s-dimensional densities (s-densities) of the set A at the point x. If D^{s}(A, x) and D^{s}(A, x) are equal, we denote their common value by D^{s}(A, x), and call it the s-dimensional density (s-density) of the set A at the point x.

Notice that (2r)^{s} = H_{δ}^{s}(B(x, r)), if δ ≥ 2r.

In the case s = n, these are the usual Lebesgue-densities. The s-densities are Borel-functions. The density-distribution in a set is obviously itself a fundamental geometric property of the set; it is moreover the basis of a number of other fundamental geometric properties.

A useful generalization of the above concept is in the following:

Definition 3.2. Whenever E ⊂ IR^{n}, x ∈ IR^{n} and h > 0, write

x + hE = {x + hy : y ∈ E},

and define

D^{s}(A, E)(x) := lim sup_{h→0+} H^{s}(A ∩ (x + hE)) ; (h diam(E))^{s},

D^{s}(A, E)(x) := lim inf_{h→0+} H^{s}(A ∩ (x + hE)) ; (h diam(E))^{s}.

We call this limit the upper, and lower s-dimensional E-density of A at x, respectively.

Notice that (h diam(E))^{s} = H_{δ}^{s}(hE) for all δ ≥ h diam(E), and H^{s}(A ∩ (x + hE)) may well be greater than H_{δ}^{s}(hE).

By studying E-densities when E is an appropriate set other than B(0, 1), one can acquire more information about the local structure of A. For example, if E is some kind of sector or angle, then E-densities reflect some directional properties of A. Indeed, such ‘angular’ densities have been used succesfully to define measure-theoretical concepts of tangents. (See e.g., in Besicovitch [2] .) We do not make further use of this definition.

If s = n, then H^{s} = (2^{n}/α^{n})λ^{n}. Thus, by the Lebesgue density theorem,

D^{s}(A, B(0, 1)) = D^{s}(A, B(0, 1)) = χ_{A} H^{s}-a.e.

if A is H^{s}-measurable and χ_{A} denotes the characteristic function of A.

If s < n, the study of densities becomes much more complicated. We prove the fundamental density theorem for Hausdorff measures. It is weaker than the Lebesgue density theorem but often has similar applications.

3.2 Elementary density bounds

In this subsection we commence our study of fundamental density properties of s-sets (and s-dimensional sets of zero measure) by proving some results which are valid for all values of s.

Theorem 3.3. Suppose H^{s}(A) < ∞, then

(i) 2^{−s} ≤ D^{s}(A, x) ≤ 1 for H^{s}-a.a. x ∈ A,

(ii) if, in addition, A is H^{s}-measurable, then

D^{s}(A, x) = 0 for H^{s}-a.a. x ∈ IR^{n} \ A.

Proof. Let us first prove the left-hand inequality of (i). The set of those x ∈ A for which D^{s}(A, x) < 2^{−s} is the union of the sets

B_{k} := {x ∈ A : H^{s}(A ∩ B(x, r)) < k ; k + 1r^{s}, 0 < r < 1 ; k} , k = 1, 2, …

It suffices to show that H^{s}(B_{k}) = 0 for every k. Fix k and let ε > 0. We can cover B_{k} with the sets E_{1}, E_{2}, … such that 0 < diam(E_{i}) < 1 ; k, B_{k} ∩ E_{i} ≠ ∅, and

∑_{i} diam(E_{i})^{s} ≤ H^{s}(B_{k}) + ε.

For each i pick an x_{i} ∈ B_{k} ∩ E_{i} and let r_{i} := diam(E_{i}). Then B_{k} ∩ E_{i} ⊆ A ∩ B(x_{i}, r_{i}) and

H^{s}(B_{k}) ≤ ∑_{i} H^{s}(B_{k} ∩ E_{i}) ≤ ∑_{i} H^{s}(A ∩ B(x_{i}, r_{i}))

< ∑_{i} k ; k + 1r_{i}^{s} = k ; k + 1 ∑_{i} diam(E_{i})^{s} ≤ k ; k + 1(H^{s}(B_{k}) + ε).  (2)

Letting ε → 0, we get H^{s}(B_{k}) ≤ k ; k+1H^{s}(B_{k}), and since H^{s}(B_{k}) < ∞, we have H^{s}(B_{k}) = 0.

To prove the right-hand inequality in (i) first observe that we may assume A to be a Borel-set because H^{s} is Borel-regular. As above, if t > 1 we set

B := {x ∈ A : D^{s}(A, x) > t},

it is enough to show H^{s}(B) = 0. Let ε > 0 and δ > 0. Again because of the regularity and since B is measurable, we find an open set U such that B ⊆ U and H^{s}(A ∩ U) < H^{s}(B) + ε. For every x ∈ B there are arbitrarily small numbers r such that 0 < r < δ/2, B(x, r) ⊂ U and

H^{s}(A ∩ B(x, r)) > t(2r)^{s}.

By the Vitali covering theorem there are disjoint balls B_{1}, B_{2}, … for which H^{s}(B \ ⋃_{i} B_{i}) = 0. Thus

H^{s}(B) + ε > H^{s}(A ∩ U) ≥ ∑_{i} H^{s}(A ∩ B_{i}) > t ∑_{i} diam(B_{i})^{s}
≥ tH_{δ}^{s}(B ∩ ⋃_{i} B_{i}) = tH_{δ}^{s}(B).

The last step follows from H_{δ}^{s}(B \ ⋃_{i} B_{i}) = 0 and the subadditivity of H_{δ}^{s}. Letting ε → 0 and δ → 0, and using t > 1 we deduce H^{s}(B) = 0.

To prove (ii) we show that for any t > 0 the set

B := {x ∈ IR^{n} \ A : D^{s}(A, x) > t}

has zero H^{s}-measure. Let ε > 0. Since H^{s}(A ∩ B) = 0, and H^{s} is Borelregular, we find an open set U such that B ⊂ U and H^{s}(A ∩ U) < ε. For each x ∈ B there exists r(x) > 0 such that B(x, r(x)) ⊂ U and

H^{s}(A ∩ B(x, r(x))) > t(2r(x))^{s}.

By Lemma 2.1 there are x_{1}, x_{2}, … ∈ B such that the balls B_{i} := B(x_{i}, r(x_{i})) are disjoint and the balls 4B_{i} cover B. Then

tH_{∞}^{s}(B) ≤ t ∑_{i} diam(4B_{i})^{s} = t4^{s} ∑_{i} diam(B_{i})^{s} < 4^{s} ∑_{i} H^{s}(A ∩ B_{i})
≤ 4^{s}H^{s}(A ∩ U) < 4^{s}ε.

Letting ε → 0 we get H_{∞}^{s}(B) = 0, whence H^{s}(B) = 0.

Theorem 3.3 (ii) has a number of important consequences. The first of these is that the densities of a measurable subset of an s-set coincide with the densities of the original set at almost all points of the subset. This result is often used in a preparatory manner; thus when examining a subset with a certain density property we can discard the remainder of the set leaving the density property holding almost everywhere.

Corollary 3.4. (Theorem on the permanence of density) Let A and B be H^{s}-measurable subsets of IR^{n} with B ⊂ A and H^{s}(A) < ∞. Then for H^{s}-almost all x ∈ B,

D^{s}(B, x) = D^{s}(A, x) and D^{s}(B, x) = D^{s}(A, x).

Proof. Writing C := A \ B we have from (ii) that D^{s}(C, x) = 0 at almost all x in B. For such x

D^{s}(A, x) = D^{s}(B, x) + D^{s}(C, x) = D^{s}(B, x) and
D^{s}(A, x) = D^{s}(B, x) + D^{s}(C, x) = D^{s}(B, x).

Remarks. Theorem 3.3 is essentially due to Besicovitch [1]. Because of it the upper densities are in general more useful than the lower densities. It may well happen that a compact set with positive and finite H^{s}-measure has zero lower density at every point.

The upper bound in 3.3 (i) is always sharp and the lower bound 2^{−s} is the best possible for s ≤ 1. It might not be the best possible for s > 1; in fact, it is not known whether 2^{−s} could always be replaced with 1 2.

Theorem 3.3 holds also if the densities are defined in terms of H_{∞}^{s} in place of H^{s}. Of course, this is an improvement only in the case of the lower bound 2^{−s}. The proof is the same.

We shall now give a simple proof to show that a Lipschitz image of a subset of IR^{m} has positive m-dimensional lower density almost everywhere. Such Lipschitz images will be studied in Section 4, and it will there be shown that the density exists and equals one.

Theorem 3.5. Let f : IR^{m} → IR^{n} be a Lipschitz map and A ⊆ IR^{m} be λ^{m}-measurable, then D^{m}(fA, x) > 0 for H^{m}-almost all x ∈ fA.

Proof. Without loss of generality we may assume λ^{m}(A) < ∞. Let E := fA, ε > 0, and let C be a (possibly empty) compact subset of E such that D^{m}(E, x) < ε for x ∈ C, finally let U be an open set with A ⊆ U and λ^{m}(U) < ∞. We shall show that

H^{m}(C) < cελ^{m}(U),

where c depends only on m and L := Lip(f). Then because of the Borelregularity of H^{m}

H^{m}({x ∈ E : D^{m}(E, x) = 0}) ≤ H^{m}({x ∈ E : D^{m}(E, x) < ε}) ≤ cελ^{m}(U)

for all ε > 0, and the theorem follows.

We assume L > 0. According to the Vitali covering theorem we can find disjoint balls B_{i} := B(x_{i}, r_{i}) and points y_{i}, i = 1, 2, … such that x_{i} ∈ C, y_{i} ∈ A, f(y_{i}) = x_{i},

H^{m}(E ∩ B_{i}) < ε diam(B_{i})^{m},

D_{i} := B(y_{i}, r_{i} ; L) ⊂ U, and

H^{m}(C \ ⋃_{i} B_{i}) = 0.

Obviously fD_{i} ⊆ B_{i}, whence the D_{i}’s are disjoint. Consequently

H^{m}(C) = ∑_{i} H^{m}(C ∩ B_{i}) < ε ∑_{i} diam(B_{i})^{m} ≤ cε ∑_{i} λ^{m}(D_{i})
≤ cελ^{m}(U),

as required.

The following definition was first used by Besicovitch [1] (n = 2, s = 1).

Definition 3.6. A point x ∈ A at which D^{s}(A, x) = D^{s}(A, x) = 1 is called a regular point of A; otherwise x is an irregular point. An s-set A is said to be regular if H^{s}-almost all of its points are regular, and irregular if almost all of its points are irregular.

Characterizing regular s-sets and obtaining bounds for the densities of s-sets are two important aims of this paper. There is no analogue of the Lebesgue density theorem; it is not in general true that an s-set has density 1 at almost all of its points. In fact one of the main results of the subject is that an s-set cannot be regular unless s is an integer.

4 The structure of integral-dimensional sets

Throughout this section 0 < m < n is a (variable) integer and we consider sets A having H^{m}(A) < ∞. The underlying principle is that such a set A can be decomposed into two parts, called rectifiable and unrectifiable, so that the geometric measure theoretic properties of the rectifiable part are similar to those of smooth surfaces whereas the properties of the unrectifiable part are completely opposite.

4.1 m-rectifiability and m-unrectifiability

Definition 4.1. A set A ⊂ IR^{n} is called m-rectifiable if there are Lipschitz maps f_{i} : IR^{m} → IR^{n}, i = 1, 2, …, such that

H^{m}(A \ ⋃_{i=1}^{∞} f_{i}IR^{m}) = 0.

A is m-unrectifiable if H^{m}(A ∩ B) = 0 for every m-rectifiable set B ⊂ IR^{n}.

Remarks. We have adopted the terminology of Federer [4] in an abbreviated form. Federer uses the terms countably (H^{m}, m) rectifiable and purely (H^{m}, m) unrectifiable. These concepts were first introduced by Besicovitch; he took a different starting point and used the terms regular and irregular. We will see how these two concepts overlap.

It is important to see that m-unrectifiability is not an entirely opposite concept to m-rectifiability, indeed, an m-rectifiable set of zero H^{m}-measure is also m-unrectifiable.

Examples of m-rectifiable sets are all subsets of IR^{m}(⊂ IR^{n}) or, more generally, subsets of m-dimensional C^{1}-submanifolds of IR^{n}. Rectifiable curves and their subsets are 1-rectifiable. A standard example of a 1-unrectifiable set is the product of the Cantor set C_{1/4} with itself.

We can modify the definition of rectifiability:

A is m-rectifiable if and only if there are m-dimensional embedded C^{1}- submanifolds M_{1}, M_{2}, … of IR^{n} such that

H^{m}(A \ ⋃_{i=1}^{∞} M_{i}) = 0.  (3)

The criteria of the definition follows trivially from the above since we can use local coordinate representations (restricted to appropriate subsets) for the C^{1}-submanifolds to give the necessary Lipschitz maps. The fact that the definition implies (3) is also straightforward if we use the C^{1}-approximation theorem for Lipschitz maps: viz. given any Lipschitz map f : IR^{m} → IR^{n}, and any ε > 0, we can find a C^{1}-map g : IR^{m} → IR^{n} such that λ^{m}({x : f(x) ≠ g(x)}) < ε. See Federer [4] Ch.3.

Or another (much easier) alternate: A is m-rectifiable if and only if there are subsets A_{0} ⊂ IR^{n}, E_{1}, E_{2}, … ⊆ IR^{m}, and Lipschitz maps f_{i} : E_{i} → IR^{n}, such that H^{m}(A_{0}) = 0 and

A = A_{0} ∪ ⋃_{i=1}^{∞} f_{i}(E_{i}).

We list some rather immediate consequences of the definition in the following lemma:

Lemma 4.2. (i) Every m-rectifiable set has σ-finite H^{m}-measure.

(ii) Any subset of an m-rectifiable set is m-rectifiable.

(iii) The countable union of m-rectifiable sets is m-rectifiable.

(iiii) If A is m-rectifiable, there is an m-rectifiable Borel-set B such that A ⊆ B and H^{m}(A) = H^{m}(B).

The next theorem enables us to treat the rectifiable and unrectifiable parts of a set independently.

Theorem 4.3. (decomposition theorem) If H^{m}(A) < ∞, there is an m-rectifiable Borel-set B such that A \ B is m-unrectifiable. Thus A has a decomposition into an m-rectifiable and m-unrectifiable part:

A = (A ∩ B) ∪ (A \ B).

Such a decomposition is unique up to H^{m}-null-sets.

Proof. Let

M := sup{H^{m}(A ∩ B) : B ⊂ IR^{n} is Borel and m-rectifiable}.

We can choose for every j = 1, 2, … an m-rectifiable Borel-set B_{j} such that H^{m}(A ∩ B_{j}) > M − 1/j. Then B := ⋃_{j=1}^{∞} B_{j} is the set we want, while the rest of the theorem is straightforward.

4.2 Rectifiability and measures in cones

Now we turn our attention to the study of the amount of A, in the sense of H^{m}-measures, near (n−m)-dimensional planes. We introduce some notation.

Let V ∈ G(n, n−m) be an (n−m)-plane through the origin, P_{V} : IR^{n} → V be the orthogonal projection onto V, and Q_{V} : IR^{n} → V^{⊥} be the orthogonal projection onto the orthogonal complement of V. If a ∈ IR^{n}, 0 < s < 1, and 0 < r ≤ ∞, we set

X(a, r, V, s) := {x : d(x − a, V) < s|x − a|, |x − a| ≤ r}
= {x : |Q_{V} (x − a)| < s|x − a|, |x − a| ≤ r}.

This is a kind of cone around the plane V + a. The number s measures the opening angle of this cone; in the case n − m = 1 it is the sine of this angle.

Now we state a lemma which is simple but very important.

Lemma 4.4. Suppose A ⊂ IR^{n}, V ∈ G(n, n − m), 0 < s < 1, and 0 < r ≤ ∞. If

A ∩ X(a, r, V, s) = ∅ for all a ∈ A,

then A is m-rectifiable.

Proof. Since A is a countable union of sets whose diameter is less than r, we may assume diam(A) < r. Clearly |Q_{V} a − Q_{V} b| ≥ s|a − b| for every a, b ∈ A, hence Q_{V} |A is one-to-one with Lipschitz inverse f = (Q_{V} |A)^{−1}, Lip(f) ≤ 1/s. Since Q_{V} A lies on an m-plane and A = f(Q_{V} A), A is mrectifiable.

Next we take a more essential step further replacing the emptiness of the intersection by a bound on its measure.

Lemma 4.5. Let V ∈ G(n, n − m), 0 < s < 1, 0 < δ < ∞, and 0 < λ < ∞. If A is m-unrectifiable and

H^{m}(A ∩ X(a, r, V, s)) ≤ λr^{m}s^{m} for all a ∈ A, 0 < r < δ,

then

H^{m}(A ∩ B(x, δ/6)) ≤ cλδ^{m} for all x ∈ IR^{n},

where c is a constant depending only on m.

Proof. Fix an x ∈ IR^{n}, we may assume A ⊂ B(x, δ ; 6) and

A ∩ X(a, ∞, V, s ; 4) ≠ ∅ for all a ∈ A,

because the subset of A where this fails has zero H^{m}-measure by Lemma 4.4. Let h : A → IR_{+},

h(a) := sup {|b − a| : b ∈ A ∩ X(a, ∞, V, s ; 4)} .

Clearly 0 < h(a) < δ ; 3. For any fixed a choose

a^{∗} ∈ A ∩ X(a, ∞, V, s ; 4) with |a − a^{∗}| ≥ 3 4h(a).

Denoting C_{a} := Q_{V}^{−1}(Q_{V} B(a, s ; 4h(a))), that is, a multi-dimensional cylinder of radius s ; 4h(a) around the plane x + V, we have

A ∩ C_{a} ⊆ X(a, 2h(a), V, s) ∪ X(a^{∗}, 2h(a), V, s), a ∈ A.  (4)

Let c ∈ A ∩ C_{a}. Since c ∈ C_{a}, we have

|Q_{V} (a − c)| ≤ s ; 4h(a).

If h(a) < |a − c|, substitute and get |Q_{V} (a − c)| < s ; 4|a − c| which means c ∈ X(a, ∞, V, s ; 4) and, by the definition of h(a), |a − c| ≤ h(a). So h(a) < |a − c| is impossible, thus we have |a − c| ≤ h(a), and |a^{∗} − c| ≤ 2h(a). Suppose z ∉ X(a^{∗}, 2h(a), V, s), then

s|a^{∗} − c| ≤ |Q_{V} (a^{∗} − z)| ≤ |Q_{V} (a^{∗} − a)| + |Q_{V} (a − c)|
< s ; 4|a − a^{∗}| + s ; 4h(a) ≤ s ; 2h(a).

Cancelling with s and using |a − a^{∗}| ≥ 3 4h(a), we get

|a − c| ≥ |a − a^{∗}| − |a^{∗} − c| > 3 4h(a) − 1 2h(a) = 1 4h(a) ≥ |Q_{V} (a − c)| ; s.

Hence c ∈ X(a, h(a), V, s) ⊂ X(a, 2h(a), V, s), which proves (4).

From (4) and the hypothesis we get H^{m}(A ∩ C_{a}) ≤ 2λ(2h(a))^{m}s^{m}. By Lemma 2.1 there is a finite or countable set {a_{1}, a_{2}, …} ⊆ A such that the balls Q_{V} B(a_{i}, s ; 16h(a_{i})), i = 1, 2, … in the m-dimensional plane V^{⊥} are disjoint and

Q_{V} A ⊆ ⋃_{i} Q_{V} B(x, s ; 4h(a_{i})),

that is, A ⊂ ⋃_{i} C_{a_{i}}. Hence (recall H^{m}(V^{⊥} ∩ B(y, r)) = 2^{m}r^{m} for y ∈ V^{⊥})

H^{m}(A) ≤ ∑_{i} H^{m}(A ∩ C_{a_{i}}) ≤ 2λ2^{m} ∑_{i}(sh(a_{i}))^{m}

= 2^{m+1}λ16^{m}2^{−m} ∑_{i} H^{m}(V^{⊥} ∩ B(Q_{V} a_{i}, s ; 20h(a_{i})))

≤ 2 · 16^{m}λH^{m}(V^{⊥} ∩ B(Q_{V} a, δ ; 2)) ≤ 2 · 16^{m}λδ^{m},

which is the asserted inequality.

The following corollary tells that an unrectifiable set is rather scattered; it approaches almost all of its points from all directions.

Corollary 4.6. If δ > 0, H^{m}(A) < ∞, and A is m-unrectifiable, then

lim sup_{s→0+} sup_{0<r<δ} H^{m}(A ∩ X(a, r, V, s)) ; (rs)^{m} > 0

for H^{m}-a.a. a ∈ A.

Proof. Let B be the set of those points a ∈ A for which the assertion fails, let λ > 0, and define

A_{i} := {a ∈ A : sup_{0<r<δ} H^{m}(A ∩ X(a, r, V, s)) ; (rs)^{m} < δ for 0 < s < 1 ; i} .

Then A_{1} ⊆ A_{2} ⊆ …, B ⊆ ⋃_{i=1}^{∞} A_{i} and, by Lemma 4.5,

H^{m}(A_{i} ∩ B(a, δ ; 6)) ≤ cλδ^{m}

for all a ∈ IR^{n}. Hence

H^{m}(B ∩ B(a, δ ; 6)) ≤ cλδ^{m}.

Letting λ → 0+ we see that B intersects every ball of radius δ/6 in a set of H^{m}-measure zero, which yields H^{m}(B) = 0.

4.3 Linear approximation properties

In this subsection we shall formulate approximation properties but only to be used as useful technical tools later on.

Definition 4.7. A subset A of IR^{n} is called m-linearly approximable if for H^{m}-almost all a ∈ A the following is true: for any (arbitrarily small) positive number ε, there are positive numbers r_{0} and β and an affine m-plane W ∈ A(n, m) such that a ∈ W and for every 0 < r < r_{0},

H^{m}(A ∩ B(b, εr)) ≥ βr^{m} for b ∈ W ∩ B(a, r),  (5)

and

H^{m}(A ∩ B(a, r) \ W (εr)) < εr^{m}.  (6)

In the weaker form of this property W is allowed to depend on r:

Definition 4.8. A subset A of IR^{n} is weakly m-linearly approximable if for H^{m}-almost all a ∈ A the following is true: for any positive number ε, there are positive numbers r_{0} and β such that for every 0 < r < r_{0} there is a W ∈ A(n, m), a ∈ W, such that (5) and (6) hold.

Definition 4.7 describes the tangent plane in a measure-theoretic sense. For an affine m-plane W to be a tangent plane for a set A at a point a means that W ∩ B(a, r) should approximate A ∩ B(a, r) reasonably well for small r. According to the condition (6) most of A lies near W in B(a, r), while (5) says that there are no big holes in A near W ∩ B(a, r).

Clearly (5) imply D^{m}(A, a) > 0 for H^{m}-almost all a ∈ A. Notice also, that if H^{m}(A) < ∞, both approximation properties are preserved for H^{m}- measurable subsets in consequence of Theorem 3.3 (ii).

Theorem 4.9. If A is an H^{m}-measurable m-rectifiable subset of IR^{n} with H^{m}(A) < ∞, then A is m-linearly approximable.

Proof. We have to show that for H^{m}-almost all points of A and for a given ε > 0, there are r_{0}, β > 0 and W ∈ A(n, m) with the prescribed properties. Let f : IR^{m} → IR^{n} be a Lipschitz map, M be a λ^{m}-measurable subset of IR^{m} with fM ⊆ A. It is enough to verify the properties (5) and (6) at H^{m}-almost all points of fM.

By Theorem 3.5, D^{m}(fM, a) > 0 for H^{m}-almost all a ∈ fM. Hence we may assume that there are r_{1} > 0 and λ > 0 such that

H^{m}(A ∩ B(a, r)) ≥ λr^{m} for a ∈ fM, 0 < r < r_{1};  (7)

because the original fM is, to within a set of measure zero, a countable union of such subsets, and we may consider each of them seperately.

If f is differentiable at a point x, define L_{x} : IR^{m} → IR^{n} : y ↦ f′(x)y − f′(x)x + f(x) and write W_{x} := L_{x}IR^{m}. Further, if dim W_{x} = m, let l(x) be the greatest number for which |L_{x}y−L_{x}x| ≥ l(x)|y−x| for all y ∈ IR^{m}. By Theorem 2.3 and since a Lipschitz function is differentiable almost everywhere (recall Rademacher’s Theorem), we may assume that l(x) > 0 exists on M.

Let 0 < ε < 2 be fixed (clearly, the greater ε is, the rather (5) and (6) hold), and denote L := Lip(f). If η is arbitrary, using Lebesgue’s density theorem for Radon measures we can find a compact subset K of M and positive numbers δ < min{ε/3, 1/L} and r_{2} such that λ^{m}(B \ C) < η and that for any x ∈ K,

2δ ≤ l(x),  (8)
|f(y) − L_{x}y| < δ^{2}|x − y| for all y ∈ B(x, r_{2}),  (9)
d(y, B) < δ^{2}r for y ∈ B(x, r ; δ), 0 < r < r_{2}.  (10)

Let r_{3} := min{r_{1}, r_{2}}, we divide K into finitely many Borel-subsets K_{i} such that diam(K_{i}) < r_{3}. Fix i and let x ∈ K_{i}, a = f(x) be such that D^{m}(A \ fK_{i}, a) = 0. By Theorem 3.3 (ii) it suffices to verify (5) and (6) at such a point a.

Let 0 < r < r_{0} := δr_{3} and b ∈ W_{x} ∩ B(a, r), b = L_{x}y. Then by (8), y ∈ B(x, r ; 2δ). By (10) there is a z ∈ B with |y − z| < δ^{2}r, whence

|x − z| < |x − y| + |y − z| < r ; 2δ + δ^{2}r < r ; 2δ + r ; 2δ = r ; δ < r_{3}. Then by (9) and

the fact that ∥f′(x)∥ ≤ L < 1/δ,

|f(z) − b| ≤ |f(z) − L_{x}z| + |L_{x}z − L_{x}y| < δ^{2}|z − x| + L|z − y| < 2δr.

Thus, as 3δ < ε, we have by (7)

H^{m}(A ∩ B(b, εr)) ≥ H^{m}(A ∩ B(f(z), δr)) ≥ λδ^{m}r^{m},

and so is (5) true (with β = λδ^{m}).

From (9) we obtain

f(K_{i} ∩ B(x, r ; δ)) ⊂ W_{x}(δr) ⊂ W_{x}(εr),

and from (8) and (9) using the fact that diam(K_{i}) < r_{0}

f(K_{i} \ B(x, r ; δ)) ⊂ IR^{n} \ B(a, r),

because for y ∈ K_{i} \ B(x, r ; δ)

|a − f(y)| ≥ |L_{x}x − L_{x}y| − |L_{x}y − f(y)|
≥ 2δ|x − y| − δ^{2}|x − y| ≥ δ|x − y| > r.

Thus

B(a, r) ∩ fK_{i} ⊂ W_{x}(εr).

Taking into account, that D^{m}(A \ fK_{i}, a) = 0, (6) follows.

4.4 A lemma on projections of unrectifiable sets

Note that after we have proved Theorem 4.11 the following lemma will became worthless: only sets of zero H^{m}-measure will satisfy its assumptions.

Lemma 4.10. Let A be an H^{m}-measurable subset of IR^{n} with H^{m}(A) < ∞. If A is m-unrectifiable and weakly m-linearly approximable, then H^{m}(P_{V} A) = 0 for all V ∈ G(n, m).

Proof. Let 0 < ε < 1/2 and V ∈ G(n, m). Using the weak m-linear approximation property we can find a compact subset C of A and positive numbers r_{0}, δ and η with δ < 1, η < δε for which the following holds: if a ∈ C and 0 < r < r_{0}, then

H^{m}(A ∩ B(a, r)) > δr^{m}  (11)

and there is a W ∈ A(n, m) such that a ∈ W and

C ∩ B(a, r) \ W (ηr) = ∅.  (12)

In fact, if one chooses C, r_{0} and δ so that for a ∈ C and 0 < r < r_{o} (11) holds (using only the positiveness of the lower density of A), and by (6) that

H^{m}(A ∩ B(a, 2r) \ W (η ; 2r)) < δ(η ; 2r)^{m},

then for b ∈ B(a, r) \ W (ηr) B(b, η ; 2r) ⊂ B(a, 2r) \ W ( η ; 2r), and (12) follows from (11). Then we also have

H^{m}(P_{V} (A \ C)) < ε.  (13)

Since C is m-unrectifiable, it follows from Lemma 4.4 that

H^{m}( ⋃_{i=1}^{∞} {a ∈ C : C ∩ X(a, 1 ; i, V^{⊥}, η) = ∅} ) = 0.

Hence for H^{m}-almost all a ∈ C there are points b ∈ C arbitrarily close to a such that

|P_{V} (b − a)| < η|b − a|.  (14)

Suppose a, b ∈ C satisfy (14) and r := |a − b| < r_{0}. Let W be as in (12) with a ∈ W and let c := P_{W} b. Using (12) we see that

|c − b| ≤ ηr, r ; 2 ≤ |c − a| ≤ r, |P_{V} (c − a)| < 2ηr.

We can select an orthonormal basis {e_{1}, …, e_{m}} for W − a so that P_{V} (e_{i}) · P_{V} (e_{j}) = 0 for i ≠ j. Then for some i,

|P_{V} e_{i}| ≤ 2 ; r |P_{V} (c − a)| < 4η,

because otherwise we would have

|P_{V} (c − a)|^{2} = ∑_{j=1}^{m} |(c − a) · e_{j}|^{2} |P_{V} e_{j}|^{2}

> 4 ; r^{2} |P_{V} (c − a)|^{2} |c − a|^{2} ≥ |P_{V} (c − a)|^{2}.

It follows that P_{V} (W ∩ B(a, r)) is contained in an m-dimensional rectangle with one side of length 8ηr and the others of length 2r. Hence by condition (12), P_{V} (C ∩ B(a, r)) is contained in a rectangle with sidelenghts 10ηr, 2r + 2ηr, …, 2r + 2ηr. Therefore, as η < 1/2 and H^{m} = (2^{m}/α^{m})λ^{m} on IR^{m}, we have with c := 20^{m}/α^{m},

H^{m}(P_{V} (C ∩ B(a, r))) ≤ cηr^{m}.  (15)

Now using the Vitali covering theorem we find disjoint balls B(a_{i}, r_{i}) satisfying (15) and such that a_{i} ∈ C and

H^{m}(C \ ⋃_{i} B(a_{i}, r_{i})) = 0.

Hence from (15) and (11) we obtain

H^{m}(P_{V} (C)) ≤ ∑_{i} H^{m}(P_{V} (C ∩ B(a_{i}, r_{i})))
≤ cη ∑_{i} r_{i}^{m} ≤ cη ; δ ∑_{i} H^{m}(A ∩ B(a_{i}, r_{i}))
≤ cηH^{m}(A).

Combining with (13), H^{m}(P_{V} (A)) < (1+cH^{m}(A))ε, and the lemma follows.

4.5 Rectifiability, densities, and projections

Theorem 4.11. Let A be an H^{m}-measurable subset of IR^{n} with H^{m}(A) < ∞. Then A is m-rectifiable if and only if A is weakly m-linearly approximable. Moreover, in this case, (i) D^{m}(A, x) = 1 for H^{m}-almost all a ∈ A, and

(ii) H^{m}(P_{V} A) > 0 for γ_{n,m}-almost all V ∈ G(n, m).

Here γ_{n,m} is the Haar measure on G(n,m).

As the proof is rather long and full of technical subtleties, first we sketch the basic ideas. The main part is in showing that the weak m-linear approximation property implies rectifiability, and the refinement of the argument gives the rest. In view of Lemma 4.10 it suffices to show that some projection of A has positive H^{m}-measure. Let C be a compact subset of A where 0 < cr^{m} ≤ H^{m}(A ∩ B(a, r)) ≤ dr^{m} < ∞ for 0 < r < r_{0} and where the approximation conditions (5) and (6) hold uniformly. Consider a ball B(a, r) such that H^{m}((A \ C) ∩ B(a, r))/r^{m} is small and most of A ∩ B(a, r) lies close to an m-plane W. Suppose that the projection of A ∩ B(a, r) on W is small. This implies that we may find many disjoint open cylinders Y_{i} of radii ϱ_{i} ≪ r orthogonal to W such that the cylinders with the same centres and radii 5ϱ_{i} are also disjoint, C ∩ B(a, r) ∩ Y_{i} = ∅, and C ∩ B(a, r) ∩ ∂Y_{i} contains a point, say e_{i}. Then for a large number M, A ∩ B(e_{i}, Mϱ_{i}) is well approximated by an m-plane W_{i}. For most of the indices i there is very little of A in B(a, r) ∩ Y_{i}, whence W_{i} must be almost orthogonal to W. This will provide us with so many disjoint balls B(x_{i,j}, ϱ_{i}) ⊂ B(a, r), x_{i,j} ∈ C that H^{m}(A ∩ ⋃_{i,j} B(x_{i,j}, ϱ_{i})) will be much greater than r^{m}, which is a contradiciton.

Now we give the detailed proof.

Proof. If A is m-rectifiable A is m-linearly approximable (whence also weakly) by Theorem 4.9. Assume that A is weakly m-linearly approximable. We have to show that A is m-rectifiable and (i) and (ii) hold.

Let ε > 0. Since A has positive lower density H^{m}-almost everywhere on A, there are positive numbers δ and r_{0}, and a compact subset C of A with H^{m}(A \ C) < ε such that

H^{m}(A ∩ B(a, r)) > δr^{m} for a ∈ C, 0 < r < r_{0}.  (16)

Let 1 2 < u < 1, 0 < γ ≤ 1, 0 < η < γ ; 8(1 − u). By Theorem 3.3 (ii) we may find a compact subset C_{1} of C and a positive number r_{1} ≤ r_{0} such that H^{m}(C \ C_{1}) < ε and for any a ∈ C, 0 < r < r_{1} there is W ∈ A(n, m) for which a ∈ W and

C_{1} ∩ B(a, 2r) \ W (ηr) = ∅  (17)
W ∩ B(a, r) ⊂ C(ηr).  (18)

Notice that when η is fixed, λ = λ(a) in Definition 4.7 depends only on a, and we can first take a large subset C′ of C such that λ(a) ≥ λ_{0} > 0 for a ∈ C′, and then apply D^{m}(A \ C, x) = 0 for H^{m}-almost all x ∈ C to find C_{1} ⊆ C′ so that (18) holds with C in place of A. By Theorem 3.3 for H^{m}-almost all a ∈ C_{1}, D^{m}(A, a) ≤ 1 and D^{m}(A \ C_{1}, a) = 0. Hence, as before, we may now deduce that for H^{m}-almost all a ∈ C_{1} there exists a positive number r_{2} ≤ r_{1} such that for 0 < r < r_{2} there is an affine subspace W ∈ A(n, m) with a ∈ W for which

C_{1} ∩ B(a, r) \ W (ηr) = ∅  (19)
W ∩ B(a, r) ⊂ C_{1}(ηr),  (20)
H^{m}(A ∩ B(a, r)) < 3^{m}r^{m},  (21)
H^{m}((A \ C_{1}) ∩ B(a, r)) < t( r ; 320)^{m},  (22)

with t = 2^{m}γ^{m}(u^{m} − u^{2m}).

Fix such an a, r, and W; and let V ∈ G(n, m) be such that P_{V} |W : W → V is one-to-one and γ ≤ ∥P_{V} |(W − a)∥. Then

|P_{V} x − P_{V} y| ≥ γ|x − y| for x, y ∈ W.  (23)

We shall show that whenever η is sufficiently small for given δ, u and γ, we have

H^{m}(P_{V} (A ∩ B(a, r))) ≥ (2γu^{2}r)^{m}.  (24)

This proves all the assertions. First, rectifiability follows from Lemma 4.10: were A not m-rectifiable we could apply (24) to an m-unrectifiable subset of A with positive H^{m}-measure. Second, choosing V := W − a we may take γ := 1, which gives for 0 < r < r_{2}

H^{m}(A ∩ B(a, r)) ≥ (2u^{2}r)^{m},

thus we get (i), for D^{m}(A, x) ≤ 1 H^{m}-almost everywhere. Third, (ii) follows since P_{V} |W is one-to-one for γ_{n,m}-almost all V ∈ G(n, m), hence γ_{n,m}({V : ∥P_{V} |(W − a)∥ < γ}) → 0 as γ → 0+, and this measure depends only on γ.

For assume that (24) fails, and set

K := P_{V} (C_{1} ∩ B(a, r)) and L := P_{V} (W ∩ U(a, ur)) \ K.

Then K is compact and

H^{m}(K) < (2γu^{2}r)^{m}.

By (23) V ∩ B(P_{V} a, γur) ⊆ P_{V} (W ∩ B(a, ur)), whence, recalling that t = 2^{m}γ^{m}(u^{m} − u^{2m}) and H^{m}(B(x, r)) = (2r)^{m},

H^{m}(D) ≥ tr^{m}.  (25)

We may cover L with balls B(b, ϱ) so that b ∈ L, K ∩ U(b, ϱ) = ∅ and K ∩ ∂B(b, ϱ) ≠ ∅. Applying the covering Lemma 2.1 to the balls B(b, 5ϱ) we may select a finite number of them, say B(b_{i}, 5ϱ_{i}), i = 1, …, p, so that

B(b_{i}, 5ϱ_{i}) ∩ B(b_{j}, 5ϱ_{j}) = ∅ for i ≠ j,  (26)

and the balls B(b_{i}, 20ϱ_{i}) cover L, thus by (25)

∑_{i=1}^{p} ϱ_{i}^{m} ≥ tr^{m} ; 40^{m} .  (27)

From (20) we obtain

ϱ_{i} ≤ ηr for i = 1, …, p.  (28)

Consider the sets

S_{i} := P_{V}^{−1}(B(b_{i}, ϱ ; 2)) ∩ W (γ(1 − u)r ; 4)

and rearrange them so that they contain no point of C for i = 1, …, q and they contain at least one point c_{i} of C for i = q + 1, …, p. Let b′_{i} := P_{V}^{−1}({b_{i}}) ∩ W, then we have b′_{i} ∈ B(a, ur), and by equations (23) and (28) for i = q + 1, …, we have

|a − c_{i}| ≤ |a − b′_{i}| + |b′_{i} − P_{W} c_{i}| + |P_{W} c_{i} − c_{i}|
≤ ur + |b_{i} − P_{V} (P_{W} c_{i})| ; γ + (1 − u)r ; 4
≤ ur + |b_{i} − P_{V} c_{i}| ; γ + P_{V} (c_{i} − P_{W} c_{i})| ; γ + (1 − u)r ; 4
≤ (1 + u)r ; 2 + ϱ_{i} ; 2γ ≤ (1 + u)r ; 2 + ηr ; γ.

As η < γ ; 3(1 − u), this gives

B(c_{i}, ϱ_{i} ; 4 ) ⊂ B(a, r).

Moreover

P_{V} (B(c_{i}, ϱ_{i} ; 4 )) ⊂ V ∩ U(b_{i}, ϱ_{i}) ⊂ V \ K.  (29)

It follows that

⋃_{i=q+1}^{p} A ∩ B(c_{i}, ϱ_{i} ; 4 ) ⊂ (A \ C_{1}) ∩ B(a, r).  (30)

Since the balls B(c_{i}, ϱ_{i} ; 4 ) are disjoint (v. (29) and (26)) we obtain from (16), (30), and (22) that

δ ; 4^{m} ∑_{i=q+1}^{p} ϱ_{i}^{m} < tδr^{m} ; 320^{m} ,

whence by (27)

∑_{i=1}^{q} ϱ_{i}^{m} > tr^{m} ; 80^{m} .  (31)

From now on we consider only i = 1, …, q. Recall (19) and how the balls B(b_{i}, ϱ_{i}) were chosen, then we can see that there exist points

e_{i} ∈ P_{V}^{−1}(∂B(b_{i}, ϱ_{i})) ∩ W (ηr) ∩ C_{1} for i = 1, …, q.  (32)

By (28) η^{−1}ϱ_{i} ≤ r < r_{1} and we can apply (18) to get W_{i} ∈ A(n, m) with e_{i} ∈ W_{i} so that

A_{i} := B(e_{i}, γ(1 − u)ϱ_{i} ; 16η ) ∩ W_{i} ⊂ C((1 − u)ϱ_{i} ; 16 ).  (33)

Thus b_{i} ∉ P_{V} A_{i}. Indeed, were there an x ∈ A_{i} with P_{V} x = b_{i}, by (33) we could find a point y ∈ C with |x − y| ≤ (1 − u) ϱ_{i} ; 16. Consequently, P_{V} y ∈ B(b_{i}, ϱ_{i} ; 2 ) would hold, and using (32), (28), and η < γ ; 8(1 − u) we would get

d(y, W) ≤ |y − x| + |x − e_{i}| + |d(e_{i}, W)|
< (1 − u)ϱ_{i} ; 16 + γ(1 − u)ϱ_{i} ; 16η + ηr < γ(1 − u)r ; 4,

whence y would belong to S_{i} ∩ C, which is empty.

Let I_{i} be the closed line segment with end-points b_{i} and P_{V} e_{i}. Then I_{i} ∩ ∂_{V} P_{V} (A_{i}) ≠ ∅, for b_{i} ∈ I_{i} \ P_{V} (A_{i}) and P_{V} e_{i} ∈ I_{i} ∩ P_{V} A_{i}. (Here ∂_{V} denotes the boundary relative to V.) As ∂_{V} P_{V} (A_{i}) = P_{V} (∂_{W_{i}}A_{i}) we may find a_{i} ∈ ∂_{W_{i}}A_{i} so that P_{V} a_{i} ∈ I_{i}. Let J_{i} be the closed line segment connecting e_{i} and a_{i}. Then J_{i} ⊂ A_{i} and P_{V} J_{i} ⊆ I_{i}, hence by (32)

|P_{V} x − b_{i}| ≤ ϱ_{i} for x ∈ J_{i}.  (34)

By (33) J_{i} is contained in the union of the balls B(x, ϱ_{i}), x ∈ C. Since the lenght of J_{i} is η^{−1}γ(1 − u)/16, we may choose a finite number of them, say B(x_{i,j}, ϱ_{i}), j = 1, …, k, such that

J_{i} ∩ B(x_{i,j}, ϱ_{i}) ≠ ∅,  (35)
B(x_{i,j}, ϱ_{i}) ∩ B(x_{i,l}, ϱ_{i}) = ∅, for j ≠ l,  (36)
k > γ(1 − u) ; 160η.  (37)

One might deduce it easily (e.g., with the help of Lemma 2.1).

Set

B_{i} := ⋃_{j=1}^{k} B(x_{i,j}, ϱ_{i}) for i = 1, …, q.

It follows from (34) and (35) that P_{V} B_{i} ⊂ B(b_{i}, 3ϱ_{i}), whence by (26) the sets B_{i} are disjoint. By (35), (28), and η < (1 − u)/8,

|x_{i,j} − e_{i}| ≤ H^{1}(J_{i}) + ϱ_{i} = γ(1 − u)ϱ_{i} ; 16η + ϱ_{i} < r ; 4(1 − u).

With b′_{i} ∈ B(a, ur) ∩ W, P_{V} b′_{i} = b_{i}, as before, (32), (23), and (28) give

|e_{i} − b′_{i}| ≤ |e_{i} − P_{W} e_{i}| + |P_{W} e_{i} − b′_{i}| ≤ ηr + |P_{V} (P_{W} e_{i}) − b_{i}| ; γ
≤ ηr + |P_{V} (P_{W} e_{i} − e_{i})| ; γ + |P_{V} e_{i} − b_{i}| ; γ
≤ ηr + ηr ; γ + ϱ_{i} ; γ ≤ 3ηr ; γ < (1 − u)r ; 2.

It follows that B_{i} ⊂ B(a, r). Using (36), (16), and (37) we obtain

H^{m}(A ∩ B_{i}) = ∑_{j=1}^{k} H^{m}(A ∩ B(x_{i,j}, ϱ_{i}))

> kδϱ_{i}^{m} > γ(1 − u)δϱ_{i}^{m} ; 160η for i = 1, …, q,

whence by (21) and (31)

3^{m}r^{m} > H^{m}(A ∩ B(a, r)) ≥ ∑_{i=1}^{q} H^{m}(A ∩ B_{i})

≥ γ(1 − u)δ ; 160η ∑_{i=1}^{q} ϱ_{i}^{m} > γ(1 − u)δtr^{m} ; 160^{m} · 80^{m}η .

This is contradictory to the fact that for given δ, u and γ we may choose arbitrarily small η.

4.6 Two equivalence theorems on rectifiability

In this subsection we only state some theorems without proving them. The first two are strongly related to Theorem 4.11. Then we discuss the sufficiency of the weaker condition that the density D^{m}(A, x) exists for H^{m}-almost all x ∈ A. We shall only prove this for m = 1, in the next subsection. The proof for general dimensions m ≥ 2 due to Preiss [11] is very complicated.

Theorem 4.12. Let A be H^{m}-measurable and H^{m}(A) < ∞. Then

(i) A is m-rectifiable if and only if the density D^{m}(A, x) exists and equals one for H^{m}-a.a. x ∈ A.

(ii) A is m-unrectifiable if and only if D^{m}(A, x) < 1 for H^{m}-a.a. x ∈ A.

First notice that according to Theorem 3.3 the two statements are equivalent. Actually we have already proved the ‘only if’ part of (i) (that is to say, the ‘if’ part of (ii)). The proof for the ‘if’ part of (i) (and the ‘only if’ part of (ii)) involves operation with the rectifiability of Radon-measures, and we do not present it here.

Corollary 4.13. Let A be an H^{m}-measurable set with H^{m}(A) < ∞. Then A is m-rectifiable if and only if A is regular, and, similarly, A is m-unrectifi-able if and only if A irregular.

Remarks. For m = 1 we know another condition: A is 1-rectifiable if and only if 0 < D^{1}(A, x) for H^{1}-a.a. x ∈ A, and A is 1-unrectifiable if and only if D^{1}(A, x) < D^{1}(A, x) for H^{1}-a.a. x ∈ A. It is not known whether this holds for m ≥ 2.

We evidently have the fact that the converse of statement (ii) of Theorem 4.11 is not enough in itself to imply rectifiability. Still we have the following theorem:

Theorem 4.14. Let A be H^{m}-measurable and H^{m}(A) < ∞. Then

(i) A is m-rectifiable if and only if H^{m}(P_{V} B) > 0 for γ_{n,m}-a.a. V ∈ G(n, m), all B ⊂ A such that B is H^{m}-measurable and H^{m}(B) > 0,

(ii) A is m-unrectifiable if and only if H^{m}(P_{V} A) = 0 for γ_{n,m}-a.a. V ∈ G(n, m).

Here too, the discussion after the previous theorem applies without change.

About (ii) we can say even more. Suppose there are V_{1}, …, V_{k} ∈ G(n, m) such that they span IR^{n} and H^{m}(P_{V_{i}}A) = 0 ∀i, then A is m-unrectifiable.

4.7 Density and the characterization of regular 1-sets

We now restrict our attention to subsets of the plane. In this subsection we derive an essential upper bound for the lower densities of an irregular 1-set and hence characterize regular and irregular 1-sets.

Lemma 4.15. A continuum (a compact connected set) A with H^{1}(A) < ∞ is arcwise connected.

This topological result is not so hard to see, one might use arcwise connectivity as an intermediate stage. A compact arcwise connected set is easily seen to be a continuum. This lemma requires the converse of it.

Lemma 4.16. Any compact arcwise connected set A with H^{1}(A) < ∞ consists of a countable union of rectifiable curves, together with a set of measure zero, or, in our terminology, is 1-rectifiable .

Proof. We define a sequence of curves {Γ_{j}} inductively. Let {Γ_{1}} be a curve in A joining two of the most distant points of A. Suppose that the curves Γ_{1}, Γ_{2}, …, Γ_{k} have been defined. Let x be a point of A at maximum distance, d_{k} say, from ⋃_{1}^{k} Γ_{j} (this maximum is attained as the sets involved are compact). If d_{k} = 0 the process terminates and the result follows. Otherwise let Γ_{k+1} be a curve in A joining x to ⋃_{1}^{k} Γ_{j} with Γ_{k+1} disjoint from ⋃_{1}^{k} Γ_{j} except for an endpoint. Then evidently

H^{1}(Γ_{k+1}) ≥ d_{k}.  (38)

Then

∑_{j=1}^{∞} d_{j} ≤ ∑_{j=1}^{∞} H^{1}(Γ_{j}) ≤ H^{1}(A) < ∞,  (39)

so that d_{j} → 0. If x ∈ A and the distance d from x to ⋃_{1}^{∞} Γ_{j} is positive, then d_{j} < d for some j, and x would have been chosen as the endpoint of Γ_{j+1}, a contradiction. We conclude that A is the closure of ⋃_{1}^{∞} Γ_{j}. Then λ^{1}(Γ_{j}) = H^{1}(Γ_{j}) ≤ H^{1}(A) < ∞, so that Γ_{j} is a rectifiable curve for each j.

Finally, for each value of k, ⋃_{1}^{k} Γ_{j} is closed, so

V := {B(x, r) : x ∈ A \ ⋃_{1}^{k} Γ_{j} and B(x, r) ∩ ⋃_{1}^{k} Γ_{j} = 0}

is such a class of balls that satisfy the conditions of the Vitali covering theorem for A \ ⋃_{1}^{k} Γ_{j}. Let x be the centre of some B ∈ V. Since x lies in the closure of ⋃_{k+1}^{∞} Γ_{j} there are points of this union arbitrarily close to x. Such points must be connected via a sequence of arcs in ⋃_{k+1}^{∞} Γ_{j} to a point of ⋃_{1}^{k} Γ_{j} necessarily outside B, so

1 2 diam(B) ≤ H^{1}(B ∩ ⋃_{k+1}^{∞} Γ_{j})

for any B ∈ V. Thus, given ε > 0, we may use the Vitali covering theorem to choose a disjoint sequence of balls {B_{i}} from V such that

H^{1}(A \ ⋃_{1}^{k} Γ_{j}) ≤ ∑_{i} diam(B_{i}) + ε ≤ 2 ∑ H^{1}(B_{i} ∩ ⋃_{k+1}^{∞} Γ_{j}) + ε

≤ 2H^{1}( ⋃_{k+1}^{∞} Γ_{j}) + ε.

Hence

H^{1}(A \ ⋃_{1}^{k} Γ_{j}) ≤ 2H^{1}( ⋃_{k+1}^{∞} Γ_{j}).

By (39) the right-hand side of this inequality tends to zero as k → ∞ so that H^{1}(A \ ⋃_{1}^{∞} Γ_{j}) = 0, as required.

Theorem 4.17. Let A be a continuum with H^{1}(A) < ∞. Then A is 1rectifiable .

Proof. Combine Lemmas 4.15 and 4.16.

Lemma 4.18. The intersection of a 1-unrectifiable set with a continuum of finite H^{1}-measure is of measure zero.

Proof. This is immediate from Theorem 4.17 and the definition of a 1unrectifiable set.

The following part is devoted to showing that a 1-unrectifiable set has lower density strictly less than 1 almost everywhere, and so is irregular. To this end we use Besicovitch’s [2] (Section 15) idea of ‘circle-pairs’. (A circlepair is a figure formed by two discs of equal radii, each with its centre on the perimeter of the other.) The proof is complicated but rather remarkable. We first require a topological result on the removal of the interiors of discs from continua. A collection of discs is semidisjoint if no member of the collection is included in any other.

Lemma 4.19. Let A be a continuum in IR^{n}. Suppose that {B_{i}}_{1}^{∞} is a countable semidisjoint collection of closed discs each contained in A with diam B_{i} → 0. Then if Γ_{i} is the perimeter of B_{i},

C := (A \ ⋃_{i} B_{i}) ∪ ⋃_{i} Γ_{i}

is a continuum.

Proof. Note that C = (A \ ⋃_{i} intB_{i}) ∪ ⋃_{i} Γ_{i}, so to show C is closed it is enough to show that ⋃_{i} Γ_{i} ⊆ C. If x ∈ ⋃_{i} Γ_{i} \ (A \ ⋃_{i} intB_{i}) ⊆ ⋃_{i} intB_{i}, then x ∈ intB_{k}, say. Let d > 0 be the distance from x to the perimeter of B_{k}. If Γ_{j} intersects B(x, 1 2d), then not only does B_{j} meet B(x, 1 2d) but it also meets IR^{n} \ B_{k}, since B_{j} ⊈ B_{k}. Thus diam(Γ_{j}) ≥ 1 2d, so only finitely many spheres Γ_{j} can cut B(x, 1 2d). Since x ∈ ⋃_{i} Γ_{i} we conclude that for some

k, x ∈ ⋃_{i=1}^{k} Γ_{i} = ⋃_{i=1}^{k} Γ_{i} ⊆ C, as required.

To prove that C is connected, suppose that C = C_{1} ∪ C_{2}, where C_{1} and C_{2} are disjoint closed sets. Let

A_{1} := C_{1} ∪ ⋃_{{i:Γ_{i}⊆C_{1}}} B_{i}, A_{2} := C_{2} ∪ ⋃_{{i:Γ_{i}⊆C_{2}}} B_{i}.

Every Γ_{i} is contained in either A_{1} or A_{2} so that A_{1} ∪ A_{2} = A. Since the discs {B_{i}} are semidisjoint and C ∩ B_{i} ⊆ ⋃_{j} Γ_{j} for each i, it is easy to check that A_{1} ∩A_{2} = ∅. The set A_{1} is closed, for if x ∈ ⋃_{{i:Γ_{i}⊆C_{1}}} B_{i}, then x is either the limit of a sequence of points in ⋃_{1}^{k} B_{i} for some k, or else the limit point of a sequence of discs with boundaries in the closed set C_{1} and radii tending to 0. In either case x ∈ C_{1}. Similarly, A_{2} is closed, so A = A_{1} ∪ A_{2} is a disjoint decomposition of A into closed sets. Since A is connected either A_{1} or A_{2} is empty, and thus either C_{1} or C_{2} is empty.

We let R(x, y) denote the common region of the circle-pair with centres x and y, so that R(x, y) = int(B(x, |x − y|) ∩ B(y, |x − y|)). Roughly speaking, we show that if the common regions of the circle-pairs with centres in a 1-set A contain very much of A, then it is possible to join up the various components of A by curves which lie to an appreciable extent in A.

Lemma 4.20. Let A be a 1-set in IR^{2} and suppose that α > 0. Let A_{0} be a compact subset of A with H^{1}(A_{0}) > 0, such that H^{1}(A ∩ R(x_{1}, x_{2})) ≥ α|x_{1} − x_{2}| if x_{1}, x_{2} ∈ A_{0}. Then there exists a continuum H such that 0 < H^{1}(H ∩ A) ≤ H^{1}(H) < ∞.

Proof. By the usual process of uniformization we may, using Theorem 3.3 (i), find ϱ_{1} > 0 and a compact set C ⊆ A_{0} with H^{1}(C) > 0 such that

H^{1}(B(x, r) ∩ A) ≤ 2 · 2r if x ∈ C and 0 < r < ϱ_{1}.  (40)

Further, there exists a point y ∈ C and a number ϱ with 0 < ϱ < 1 10ϱ_{1} such that

H^{1}((A \ C) ∩ B(y, r)) < 2r · 10^{−3}α, 0 < r ≤ 3ϱ  (41)

and

H^{1}(C ∩ B(y, ϱ)) > 1 4 · 2ϱ = 1 2ϱ.  (42)

By reducing ϱ if necessary, we may also assume that the perimeter Γ of the disc B(y, ϱ) contains some point of C.

Let C be the family of closed discs

C := {B(x, r) : x ∈ C ∩ B(y, ϱ), 0 < r < 2ϱ
and H^{1}((A \ C) ∩ B(x, r)) ≥ αr}.  (43)

By Lemma 2.1 we may find a null, finite or countable disjoint collection of closed discs {B_{i}} in C such that ⋃_{B∈C} B ⊂ ⋃_{i} B′_{i}, where B′_{i} is the closed disc concentric with B_{i} and of four times the radius. Further, we may take the discs {B′_{i}} to be semidisjoint (by simply dropping out the superfluous ones). Write Γ_{i} for the perimeter of B′_{i}.

Letting

G := (C ∩ B(y, ϱ)) ∪ Γ ∪ ⋃_{i} B′_{i},

we define

H := (G \ ⋃_{i} B′_{i}) ∪ ⋃_{i} Γ_{i}.

The remainder of the proof, which is divided into six stages, checks that H is a continuum with the prescribed properties.

(a) G is closed: since C ∩ B(y, ϱ) and Γ are closed, it is enough to show

that ⋃_{i} B′_{i} ⊆ G. But if z ∈ ⋃_{i} B′_{i}, then either z ∈ ⋃_{i=1}^{k} B′_{i} ⊆ G for some

k < ∞, or else z is the limit of points from a subsequence of the discs {B′_{i}}_{i}. On summing areas, ∑_{i} diam(B_{i})^{2} < ∞, so diam(B′_{i}) → 0 as i → ∞, thus z is the limit point of the centres of those discs which all lie in the closed set C ∩ B(y, ϱ).

(b) G is connected: we suppose that G = G_{1} ∪ G_{2}, where G_{1} and G_{2} are non-empty disjoint closed sets, and derive a contradiction. Any connected subset of G must be contained in either G_{1} or G_{2}, so assume that Γ ⊆ G_{1}. Each B′_{i} has centre in B(y, ϱ), so if B′_{i} contains points on or outside Γ it must meet Γ and therefore be contained in G_{1}. Thus G_{2} ⊂ intB(y, ϱ).

Let G′_{1} be the set obtained by adjoining IR^{2} \ B(y, ϱ) to G_{1}, so that G′_{1} is closed, connected and disjoint from G_{2}. Since both G′_{1} and G_{2} contain points of C, let x_{1} ∈ C ∩ G′_{1} and x_{2} ∈ C ∩ G_{2} be points that minimize the distance r := |x_{1} − x_{2}|. This infimum is attained and is positive. As Γ ⊆ G′_{1} contains a point of C, 0 < r < 2ϱ. The common region R(x_{1}, x_{2}) is disjoint from C, otherwise r could be further reduced, so from the hypothesis of the lemma

0 < αr < H^{1}(A ∩ R(x_{1}, x_{2})) = H^{1}((A \ C) ∩ R(x_{1}, x_{2}))
≤ H^{1}((A \ C) ∩ B(x_{2}, r)).

Since x_{2} ∈ C ∩ B(y, ϱ) and r < 2ϱ, B(x_{2}, r) is a member of C and so B(x_{2}, r) ⊂ ⋃_{i} B′_{i} ⊂ G. Thus B(x_{2}, r) = (B(x_{2}, r) ∩ G′_{1}) ∪ (B(x_{2}, r) ∩ G_{2}) is a decomposition of B(x_{2}, r) into disjoint closed sets with x_{1} in the first and x_{2} in the second. Clearly, this is absurd, since B(x_{2}, r) is connected.

(c) H is a continuum: by (a) and (b) G is a continuum, so by Lemma 4.19 H is also a continuum.

(d) ∑_{i} diam(B′_{i}) ≤ 1 8ϱ: using (43), the disjointedness of the B_{i}, and (41), we see that

∑_{i} diam(B′_{i}) = 4 ∑_{i} diam(B_{i}) ≤ 8 ; α ∑_{i} H^{1}((A \ C) ∩ B_{i})
≤ 8 ; αH^{1}((A \ C) ∩ B(y, 3ϱ)) ≤ 8 ; α · 6ϱ10^{−3}α ≤ 1 8ϱ.

(e) H^{1}(H) < ∞: since H ⊂ A ∪ Γ ∪ ⋃_{i} Γ_{i}, we have, using (d),

H^{1}(H) ≤ H^{1}(A) + 2πϱ + π ∑_{i} diam(B′_{i}) < ∞.

(f) H^{1}(H ∩ A) > 0: from the definition of H

H^{1}(H ∩ A) ≥ H^{1}(G ∩ A) − H^{1}(A ∩ ⋃_{i} B′_{i})
≥ H^{1}(C ∩ B(y, ϱ)) − ∑_{i} H^{1}(A ∩ B′_{i})
≥ H^{1}(C ∩ B(y, ϱ)) − 2 ∑_{i} diam(B′_{i})
≥ 1 2ϱ − 1 4ϱ = 1 4ϱ > 0,

using (40) (note that diam(B′_{i}) < 20ϱ ≤ 2ϱ_{1}), with (42) and (d) providing the final estimates.

The proof of the theorem on the lower densities of 1-unrectifiable sets is now an easy consequence of the geometry of circle-pairs. We exploit the fact that a circle-pair with centres at points of high lower density must contain a subset of positive measure in its common region.

Theorem 4.21. Let A be a 1-unrectifiable set in IR^{2}. Then D^{1}(A, x) ≤ 3 4 at almost all x ∈ A.

Proof. Suppose that for some α > 0 the set A_{1} := {x : D^{1}(A, x) > 3 4 + α} has positive measure. Then we may, in the usual way, find a compact 1-set A_{2} ⊆ A_{1}, and ϱ > 0 such that

H^{1}(A ∩ B(x, r)) > (3 4 + α)2r, x ∈ A_{2}, 0 < r ≤ ϱ  (44)

and

H^{1}(A ∩ U) < (1 + α) diam(U) x ∈ A_{2} ∩ U, 0 < diam(U) ≤ 3ϱ. (45)

Let A_{0} be a compact subset of A_{2} with 0 < H^{1}(A_{0}) < ∞ and diam(A_{0}) ≤ ϱ. If x_{1}, x_{2} ∈ A_{0}, then r := |x_{1} − x_{2}| ≤ ϱ, so noting that any circle intersects A in a set of measure zero,

H^{1}(A ∩ R(x_{1}, x_{2}))

= H^{1}(A ∩ B(x_{1}, r)) + H^{1}(A ∩ B(x_{2}, r)) − H^{1}(A ∩ (B(x_{1}, r) ∪ B(x_{2}, r)))

≥ 2(3 4 + α)2r − (1 + α)3r = αr = α|x_{1} − x_{2}|,

using (44) and (45) with U := B(x_{1}, r) ∩ B(x_{2}, r). Hence applying Lemma 4.20 to A_{0} ⊆ A, we deduce that there exists a continuum H with 0 < H^{1}(H ∩ A) ≤ H^{1}(H) < ∞. Since A is a 1-unrectifiable set this is impossible by Lemma 4.18. We conclude that H^{1}(A_{1}) = 0 if α > 0 as required.

Generalization of the theory to integral-dimensional subsets of IR^{n} (n ≥ 3) was not completed until some 47 years after the publication of Besicovitch’s first fundamental paper. This was because of the considerable technical difficulties that had to be overcome; for example, the theorem that a continuum of finite H^{s}-measure is a union of rectifiable curves has no analogue if s is an integer greater than 1.

5 Structure of sets of non-integral dimension

This section examines local properties of s-sets in IR^{n} for non-integral s. The fundamental result is that any such set has lower density strictly less than 1 at almost all of its points. Indeed, the stronger result that its density fails to exist at almost all of its points has also been established. As with sets of integral dimension, higher-dimensional analogues present formidable difficulties.

5.1 s-sets with 0 < s < 1

First we consider s-sets in IR^{n} for s strictly less than 1. In this case the basic properties, including non-existence of the density almost everywhere, are relatively easy to obtain.

The following topological observation about such sets is sometimes useful.

Lemma 5.1. An s-set A in IR^{n} with 0 < s < 1 is totally disconnected.

Proof. For assume the contrary and let x and y be distinct points in the same connected component of A. Define a mapping f : IR^{n} → [0, ∞) by f(z) := |z−x|. Since f does not increase distances it follows that H^{s}(f(A)) ≤ H^{s}(A) < ∞. As s < 1 it follows that f(A) is a subset of IR of Lebesgue measure zero and, in particular, has dense complement. Choosing a number r with r ∉ f(A) and 0 < r < f(y) we have

A = {z ∈ A : |z − x| < r} ∪ {z ∈ A : |z − x| > r}.

This is an open decomposition of A with x in one part and y in the other, contradicting the assumption that x and y belong to the same connected component of A.

Theorem 5.2. If A is an s-set with 0 < s < 1, the density D^{s}(A, x) fails to exist at almost all points of A.

Proof. If the conclusion is false, A has a measurable subset of positive measure where the density exists and is at least 2^{−s} > 1 2, by Theorem 3.3 (i). Choosing ε small enough we may find an s-set C ⊆ A such that if x ∈ C, then D^{s}(A, x) exists and

H^{s}(A ∩ B(x, r)) > 1 2(2r)^{s}  (46)

for all r < ε. By regularity of H^{s} we may further assume that C is closed. Let y be an accumulation point of C and let η be a number with 0 < η < 1. If R_{r,η} denotes the annular region B(y, r(1 + η)) \ B(y, r(1 − η)), then

lim_{r→0} H^{s}(A ∩ R_{r,η}) ; (2r)^{s}
= lim_{r→0} H^{s}(A ∩ B(y, r(1 + η))) ; (2r)^{s} − H^{s}(A ∩ B(y, r(1 − η))) ; (2r)^{s}
= D^{s}(A, y)((1 + η)^{s} − (1 − η)^{s}).  (47)

On the other hand, for arbitrarily small values of r we may find x ∈ C with |x − y| = r. Then

B(x, 1 2rη) ⊂ R_{r,η},

so (46) gives

1 2r^{s}η^{s} < H^{s}(A ∩ B(x, 1 2rη)) ≤ H^{s}(A ∩ R_{r,η})

for arbitrarily small values of r. Using (47) we conclude that

2^{−(s+1)}η^{s} ≤ D^{s}(A, y)((1 + η)^{s} − (1 − η)^{s}) = D^{s}(A, y)(2sη + O(η^{2}))

as η → 0. This is impossible if s < 1 and the theorem follows from this contradiction.

Corollary 5.3. Any H^{s}-measurable set A with H^{s}(A) < ∞ is irregular.

5.2 Remarks on sets in higher dimensions

The principal higher-dimensional result is similar to Corollary 5.3.

Theorem 5.4. An H^{s}-measurable set A in IR^{n} with H^{s}(A) < ∞ is irregular unless s is an integer.

This has been proved for 0 < s < 1, and the proof is not difficult in the case n − 1 < s < n. However, for intermediate values of s, substantial new ideas were required, and the theorem was eventually proved by Marstrand [7]. Indeed, Marstrand’s work goes much farther than this, not only in that it applies to more general measures, but also in that it shows that the density fails to exists almost everywhere if s is non-integral. The proof involves a much deeper study of weak tangential properties of regular sets.

References

[1] A. S. Besicovitch: On the fundamental geometrical properties of linearly measurable plane sets of points Math. Ann. 98 (1928), 422–464.

[2] A. S. Besicovitch: On the fundamental geometrical properties of linearly measurable plane sets of points II ib. 115 (1938), 296–329.

[3] M. Chlebík: Hausdorff lower s-densities and rectifiability of sets in nspace preprint

[4] H. Federer: Geometric Measure Theory Springer-Verlag, 1969.

[5] B. Kirchheim: Rectifiable metric spaces: Local structure and regularity of the Hausdorff measure Proc. Amer. Math. Soc. 121 (1994), 113–123.

[6] J. M. Marstrand: Hausdorff two-dimensional measure in 3 space Proc. London Math. Soc. (3) 11 (1961), 91–108.

[7] J. M. Marstrand: The (ϕ, s)-regular subsets of n-space Trans. Amer. Math. Soc. 113 (1964), 369–392.

[8] P. Mattila: Hausdorff m regular and rectifiable sets in n-space Trans. Amer. Math. Soc 205 (1975), 263–274.

[9] P. Mattila: Lecture Notes on Geometric Measure Theory Universidad de Extremadura, 1986.

[10] P. Mattila: Geometry of sets and measures in Euclidean spaces Cambridge University Press, 1995.

[11] D. Preiss: Geometry of measures in IR^{n}: distribution, rectifiability, and densities Ann. of Math. 125 (1987), 537–643.

[12] D. Preiss and J. Tišer: On Besicovitch’s 1 2-problem J. London Math. Soc. (2) 45 (1992), 279–287.