ON ABSOLUTE CONTINUITY

Zoltan Buczolich Department of Analysis, Eőtvős Loránd University, Múzeum krt. 6–8, H–1088 Budapest, Hungary

Washek F. Pfeffer Department of Mathematics, University of California, Davis, CA 95616, USA

We prove that in any dimension a variational measure associated with an additive continuous function is σ-finite whenever it is absolutely continuous. The one-dimensional version of our result was obtained in [1] by a different technique. As an application, we establish a simple and transparent relationship between the Lebesgue integral and the generalized Riemann integral defined in [7, Chapter 12]. In the process, we obtain a result (Theorem 4.1) involving Hausdorff measures and Baire category, which is of independent interest. As variations defined by BV sets coincide with those defined by figures [8], we restrict our attention to figures.

The set of all real numbers is denoted by ℝ, and the ambient space of this paper is ℝ^{m} where m ≥ 1 is a fixed integer. In ℝ^{m} we use exclusively the metric induced by the maximum norm | · |. The usual inner product of x, y ∈ ℝ^{m} is denoted by x·y, and 0 denotes the zero vector of ℝ^{m}. For an x ∈ ℝ^{m} and ε > 0, we let

B_{ε}(x) = {y ∈ ℝ^{m} : |x − y| < ε} .

The closure, interior, boundary, and diameter of a set E ⊂ ℝ^{m} are denoted by cl E, int E, ∂E, and d(E), respectively.

The Lebesgue measure in ℝ^{m} is denoted by λ; however, for E ⊂ ℝ^{m}, we write |E| instead of λ(E). A set E ⊂ ℝ^{m} with |E| = 0 is called negligible. We say sets A, B ⊂ ℝ^{m} overlap if |A ∩ B| > 0. Unless specified otherwise, the words “measure” and “measurable” as well as the expressions “almost all,” “almost everywhere,” and “absolutely continuous” always refer to the Lebesgue measure λ.

The s-dimensional Hausdorff measure in ℝ^{m} is denoted by H^{s}. A set T ⊂ ℝ^{m} of σ-finite measure H^{m−1} is called thin. Thin sets will play an important role in our exposition.

Unless specified otherwise, a number is an extended real number, and a function is an extended real-valued function.

1 Charges

A cell is a compact nondegenerate subinterval of ℝ^{m}, and a figure is a finite (possibly empty) union of cells. If A and B are figures, then so are A ∪ B,

A ⊙ B = cl (int A ∩ int B) and A ⊖ B = cl (A − B) .

Each figure A has the perimeter ∥A∥ = H^{m−1}(∂A), and the unit exterior normal ν_{A} defined, in the obvious way, H^{m−1}-almost everywhere on ∂A.

The family F of all figures is topologized as follows. For n = 1, 2, …, topologize the families

F_{n} = {A ∈ F : A ⊂ B_{n}(0) and ∥A∥ < n}

by the metric ρ(A, B) = |(A − B) ∪ (B − A)|, and give F the largest topology τ for which all inclusion maps F_{n} ↪ F are continuous. It is easy to verify that the topology τ is Hausdorff, sequential [3, p. 78], but not metrizable. Moreover, τ is induced by a uniformity, and the completion of (F, τ) is the space of all bounded BV sets [4, Section 5.1].

A charge is an additive τ-continuous real-valued function defined on the family F. Explicitly, F : F → ℝ is a charge if it satisfies the following two conditions.

Additivity: F (A ∪ B) = F (A) + F (B) for each pair of nonoverlapping figures A and B.

Continuity: Given ε > 0, there is an η > 0 such that |F (C)| < ε for each figure C ⊂ B(0, 1/ε) with ∥C∥ < 1/ε and |C| < η.

To illustrate the concept of charge, we give two important examples.

1. If f is a locally integrable function on ℝ^{m}, let F (A) = ∫_{A} f dλ for each figure A. Then F is a charge by the absolute continuity of the Lebesgue integral.

2. If v is a continuous vector field on ℝ^{m}, let F (A) = ∫_{∂^{∗}A} v · ν_{A} dH^{m−1} for each figure A. Then F is a charge, called the flux of v [7, Proposition 11.2.8].

2 Derivates

Given an η > 0, we say a figure A is η-regular if

\frac{|A|}{[d(A)∥A∥} > η .

The closer η is to 1/(2m), the more cube-like are η-regular figures. In particular, no η-regular figure is empty.

Let x ∈ ℝ^{m}, and let F be a real-valued function defined on F. For a positive η < 1/(2m), let

\overline{D}_{η}F (x) = inf_{δ>0} [sup_{B} \frac{F (B)}{|B|}]

where B is an η-regular figure with x ∈ B and d(B) < δ. The number

\overline{D}F (x) = sup_{0<η< \frac{1}{2m}} \overline{D}_{η}F (x)

is called the upper derivate of F at x. Using an argument similar to [10, Chapter IV, Theorem 4.2], it is easy to show that the functions \overline{D}_{η}F and \overline{D}F, defined on ℝ^{m} in the obvious way, are measurable.

If \overline{D}F (x) = −\overline{D}(−F)(x) ≠ ±∞, we denote this common value by DF (x), and say F is derivable at x; the number DF (x) is called the derivate of F at x. When \overline{D}_{η}|F |(x) < +∞ for all positive η < 1/(2m), we say that F is almost derivable at x (cf. [7, Section 11.7]); in particular, F is almost derivable at x whenever \overline{D}|F |(x) < +∞. The term “almost derivable” is justified by the following result proved in [2, Theorem 3.3].

Theorem 2.1. Let F be a charge, and let E be the set of all x ∈ ℝ^{m} at which F is almost derivable. Then F is derivable at almost all x ∈ E.

3 Variations

A partition is a collection (possibly empty) P = {(A_{1}, x_{1}), …, (A_{p}, x_{p})} where A_{1}, …, A_{p} are nonoverlapping figures and x_{i} ∈ A_{i} for i = 1, …, p. Given η > 0 and a nonnegative function δ on E ⊂ ℝ^{m}, the partition P is called

η-regular if each figure A_{i} is η-regular,

δ-fine if x_{i} ∈ E and d(A_{i}) < δ(x_{i}) for i = 1, …, p,

in E if ⋃_{i=1}^{p} A_{i} ⊂ E.

If P is δ-fine, then each x_{i} belongs to the set {x ∈ E : δ(x) > 0}. A nonnegative real-valued function δ defined on a set E ⊂ ℝ^{m} is called a gage on E if the zero set Z_{δ} = {x ∈ E : δ(x) = 0} of δ is thin. The following proposition, established in [6], connects partitions with charges.

Proposition 3.1. Let F be a charge, and let 0 < ε < 1/(2m). For any gage δ on a figure A, there is an ε-regular δ-fine partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})} in A such that

|F (A ⊖ ⋃_{i=1}^{p} A_{i})| < ε .

Let F be a function on F, and let E ⊂ ℝ^{m}. For 0 < η < 1/(2m), let

V_{η}F (E) = inf_{δ} sup_{P} ∑_{i=1}^{p}|F (A_{i})|

where δ is a gage on E and P = {(A_{1}, x_{1}), …, (A_{p}, x_{p})} is an η-regular δ-fine partition, and

V_{∗}F (E) = sup_{0<η< \frac{1}{2m}} V_{η}F (E) .

It is easy to verify the functions V_{η}F and V_{∗}F, defined in the obvious way, are Borel regular measures in ℝ^{m} [2, Lemma 4.6] that vanish on thin sets. The measure V_{∗}F is called the critical variation of F. For charges, the critical variation V_{∗}F extends the classical variation V F [10, Chapter III, Section 4] from figures to arbitrary subsets of ℝ^{m}. We state this explicitly in the following proposition proved in [2, Lemma 4.4].

Proposition 3.2. If F is a charge, then V_{∗}F (A) = V F (A) for each figure A.

If F is a function on F and A ∈ F, we define a function F L A on F by the formula (F L A)(B) = F (A ⊙ B) for each B ∈ F. Similarly, if μ is a measure in ℝ^{m} and A ⊂ ℝ^{m}, we define a measure μ L A by the formula (μ L A)(B) = μ(A ∩ B) for each B ⊂ ℝ^{m}. Since the boundary of any figure is thin, it is easy to verify that

V_{η}(F L A) = (V_{η}F) L A  and  V_{∗}(F L A) = (V_{∗}F) L A

for each figure A. A charge in a figure A is a charge F with F = F L A. Similarly, a measure in a set E ⊂ ℝ^{m} is a measure μ with μ = μ L E.

Recall from [10, Chapter III, Section 12] a charge F in a figure A is called absolutely continuous (abbreviated as AC) if given ε > 0, there is a Δ > 0 such that

∑_{B∈B}|F (B)| < ε

for each finite collection B of nonoverlapping subfigures of A. Also recall the classical variation V F of an AC charge F is finite

Proposition 3.3. A charge F in a figure A is AC if and only if V_{∗}F is AC and finite.

Proof. Let F be AC and ε > 0. Choose a Δ > 0 as in the definition above. Given a negligible set E ⊂ A, find an open set U so that E ⊂ U and |U| < Δ. There is a positive function δ on E such that B_{δ}(x)(x) ⊂ U for each x ∈ E. If P = {(A_{1}, x_{1}), …, (A_{p}, x_{p})} is a δ-fine partition, then P is a partition in U by the choice of δ. Thus

∑_{i=1}^{p} |A ⊙ A_{i}| ≤ ∑_{i=1}^{p} |A_{i}| ≤ |U| < Δ ,

and we have

∑_{i=1}^{p}|F (A_{i})| = ∑_{i=1}^{p}|F (A ⊙ A_{i})| < ε .

This implies V_{∗}F (E) ≤ ε and, as ε is arbitrary, V_{∗}F (E) = 0. Since V_{∗}F is a measure in A, it is AC. In view of Proposition 3.2,

V_{∗}F (ℝ^{m}) = V_{∗}F (A) = V F (A) < +∞ .

Conversely, assume V_{∗}F is AC and finite. Given ε > 0, a standard argument [9, Theorem 6.11] shows there is a Δ > 0 such that V_{∗}F (E) < ε for each Borel set E with |E| < Δ. Select a finite collection B of nonoverlapping subfigures of A with | ⋃ B| = ∑_{B∈B} |B| < Δ. Proposition 3.2 implies

∑_{B∈B}|F (B)| ≤ ∑_{B∈B} V F (B) = ∑_{B∈B} V_{∗}F (B) = V_{∗}F (⋃ B) < ε .

Corollary 3.4. A charge F in a figure A is the indefinite Lebesgue integral of a function f on A if and only if V_{∗}F is AC and finite.

4 The main result and its application

A support of a measure μ in ℝ^{m} is the set

supp μ = ℝ^{m} − ⋃{G ⊂ ℝ^{m} : G is open and μ(G) = 0} .

Clearly μ(U) > 0 for each open set U ⊂ ℝ^{m} with U ∩ supp μ ≠ ∅. Since ℝ^{m} is Lindelöf, μ(ℝ^{m} − supp μ) = 0. We say E ⊂ ℝ^{m} is a supporting set if |E| > 0 and E = supp (λ L E).

In order to prove our main theorem, we need a technical result concerning Hausdorff measures. It reconciles a particular problem arising from the testy relationship between measure and category. We postpone its proof to Section 5.

Theorem 4.1. Let 0 < s < m, and let E ⊂ ℝ^{m} be a measurable set with |E| > 0. There is a supporting compact set C ⊂ E and a G_{δ} set H ⊂ C satisfying the following conditions:

1. H is dense in C and |H| = 0;

2. given Z ⊂ ℝ^{m} with H^{s}(Z) = 0, we can find a supporting compact set C_{Z} ⊂ C − Z so that H ∩ C_{Z} is dense in C_{Z}.

Proposition 4.2. Let F be a charge, and let 0 < η < 1/(2m). If V_{η}F is absolutely continuous, then

E = \{x ∈ ℝ^{m} : \overline{D}_{η}|F |(x) = +∞\}

is a negligible set.

Proof. Proceeding towards a contradiction, suppose |E| > 0. As E is measurable, there is a supporting compact set C ⊂ E and a negligible G_{δ} set H ⊂ C satisfying the conditions of Theorem 4.1 with m − 1 < s < m. Choose a gage δ on H, and use condition 2 of Theorem 4.1 to find a supporting compact set K ⊂ C − Z_{δ} so that D = H ∩ K is dense in K; this is possible as H^{s}(Z_{δ}) = 0. Applying the Baire category theorem to the G_{δ} set D, find a t > 0 and an open set U ⊂ ℝ^{m} so that D ∩ U ≠ ∅ and the set D_{t} = {x ∈ D ∩ U : δ(x) > t} is dense in D ∩ U and hence in K ∩ U.

The family B of all η-regular figures B with |F (B)| ≥ |B|/|K ∩ U| and d(B) < t is a Vitali’s cover of K ∩ U. Thus up to a negligible set, K ∩ U is covered by a disjoint sequence {B_{i}} in B. We may assume the interior of each B_{i} meets K ∩ U, and select an x_{i} ∈ B_{i} ∩ D_{t}. Now

∑_{i=1}^{∞}|F (B_{i})| > \frac{1}{|K ∩ U|} ∑_{i=1}^{∞} |B_{i}| ≥ 1 ,

and there is an integer p ≥ 1 with ∑_{i=1}^{p}|F (B_{i})| > 1. As {(B_{1}, x_{1}), …, (B_{p}, x_{p})} is an η-regular δ-fine partition, the arbitrariness of δ implies V_{η}F (H) ≥ 1, a contradiction.

Corollary 4.3. Let F be a charge. If V_{∗}F is AC, then F is derivable almost everywhere.

Proof. If V_{∗}F is AC, then so is V_{η}F for each positive η < 1/(2m). Observe

E = ⋃_{n=2m+1}^{∞}\{x ∈ ℝ^{m} : \overline{D}_{1/n}F (x) = +∞\}

is the set of all x ∈ ℝ^{m} at which F is not almost derivable. Since E is negligible by Proposition 4.2, the corollary follows from Theorem 2.1.

Theorem 4.4. Let F be a charge. If V_{∗}F is AC, then up to a thin set, ℝ^{m} is the union of compact sets K_{1}, K_{2}, … with V_{∗}F (K_{n}) < +∞ for n = 1, 2, . . .. In particular, V_{∗}F is σ-finite.

Proof. By [2, Proposition 4.2 and Corollary 4.8], the equality

V_{∗}F (E) = ∫_{E} \overline{D}|F |(x) dλ(x)

holds for each measurable set E ⊂ ℝ^{m}. Corollary 4.3 implies \overline{D}|F |(x) = |DF (x)| < +∞ almost everywhere. Thus ℝ^{m} is the union of a negligible set and the sets

E_{n,k} = \{x ∈ B_{n}(0) : \overline{D}|F |(x) ≤ k\} .

As V_{∗}F (E_{n,k}) ≤ k(2n)^{m} for k, n = 1, 2, …, the measure V_{∗}F is σ-finite.

We complete the argument by showing that, up to a thin set, each set E ⊂ ℝ^{m} with V_{∗}F (E) < +∞ can be covered by closed sets C_{1}, C_{2}, … with V_{∗}F (C_{n}) < +∞. To this end, choose a positive η < 1/(2m), and find a gage δ on E so that

∑_{i=1}^{p}|F (A_{i})| < V_{∗}F (E) + 1

for each η-regular δ-fine partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})}. For n = 1, 2, …, let E_{n} = {x ∈ E : d(x) > 1/n} and C_{n} = cl E_{n}. Since E = Z_{δ} ∪ ⋃_{n=1}^{∞} E_{n}, it remains to show V_{∗}F (C_{n}) < +∞. Define a positive function δ_{n} on C_{n} by letting

δ_{n}(x) = \begin{cases}δ(x) & if x ∈ E_{n}, \\ \frac{1}{n} & if x ∈ C_{n} − E_{n}, \end{cases}

and choose an η-regular δ_{n}-fine partition {(B_{1}, y_{1}), …, (B_{p}, y_{p})}. If y_{i} ∈ C_{n} − E_{n}, select an x_{i} ∈ E_{n} and a cube K_{i} containing x_{i}. For i = 1, ˙,p, let

(A_{i}, x_{i}) = \begin{cases}(B_{i}, y_{i}) & if y_{i} ∈ E_{n}, \\ (B_{i} ∪ K_{i}, x_{i}) & if y_{i} ∈ C_{n} − E_{n}. \end{cases}

Clearly, it is possible to choose x_{i} so close to y_{i} and K_{i} so small that {(A_{1}, x_{1}), …, (A_{p}, x_{p})} is an η-regular δ-fine partition, and

∑_{i=1}^{p}|F (B_{i})| < ∑_{i=1}^{p}|F (A_{i})| + 1 < V_{∗}F (E) + 2 .

As η is arbitrary, we conclude V_{∗}F (C_{n}) ≤ V_{∗}F (E) + 1.

We recall the definition of the generalized Riemann integral (abbreviated as R-integral) introduced in [7, Chapter 12, particularly Theorem 12.2.2].

Definition 4.5. A function f on a figure A is R-integrable whenever there is a charge F in A satisfying the following condition: given ε > 0, we can find a gage δ on A so that

∑_{i=1}^{p}|f(x_{i})|A_{i}| − F (A_{i})| < ε

for each ε-regular δ-fine partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})} in A. The charge F, which is uniquely determined by f, is called the indefinite R-integral of f.

Note that neither the R-integrability of f nor the indefinite R-integral of f depend on the values f takes on a negligible set. If f and g are R-integrable and have the same indefinite R-integral, then f = g almost everywhere.

Theorem 4.6. A charge F in a figure A is an indefinite R-integral of a function f on A if and only if V_{∗}F is AC. In which case DF = f almost everywhere in A, the function f is measurable, and V_{∗}F (E) = ∫_{E} |f| dλ for each measurable set E ⊂ A.

Proof. Suppose V_{∗}F is AC, and denote by N the negligible set of all x ∈ A at which F is not derivable (Corollary 4.3). Let

f(x) = \begin{cases}0 & if x ∈ N, \\ DF (x) & if x ∈ A − N, \end{cases}

and choose a positive ε < 1/(2m). There is a gage δ_{N} on N such that

∑_{i=1}^{p}|F (A_{i})| < ε

for each ε-regular δ_{N}-fine partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})}. Making N larger and δ_{N} smaller, we may assume ∂A ⊂ Z_{δ_{N}} and B_{δ_{N} (x)}(x) ⊂ A for each x ∈ N ∩ int A. Given x ∈ A − N, find a Δ_{x} > 0 so that B_{Δ_{x}}(x) ⊂ A, and

|\frac{F (B)}{|B|} − f(x)| < ε

for each ε-regular figure B with x ∈ B and d(B) < Δ_{x}. Define a gage δ on A by the formula

δ(x) = \begin{cases}δ_{N}(x) & if x ∈ N, \\ Δ_{x} & if x ∈ A − N. \end{cases}

Let P = {(A_{1}, x_{1}), …, (A_{p}, x_{p})} be an ε-regular δ-fine partition, and note P is a partition in A by the choice of δ. From the inequality

∑_{i=1}^{p}|f(x_{i})|A_{i}| − F (A_{i})| = ∑_{x_{i}∈N}|F (A_{i})| + ∑_{x_{i}∈A−N}|f(x_{i})|A_{i}| − F (A_{i})|

< ε + ∑_{x_{i}∈A−N} ε|A_{i}| ≤ ε(1 + |A|)

we deduce F is the indefinite R-integral of f. Since DF = f almost everywhere, f is measurable by [10, Chapter IV, Theorem 4.2]. Finally [2, Theorem 4.7] implies V_{∗}F (E) = ∫_{E} |f| for each measurable set E ⊂ A.

Conversely, suppose F is the indefinite R-integral of a function f on A, and select a negligible set N ⊂ A. With no loss of generality, we may assume f(x) = 0 for each x ∈ N. Given a positive ε < 1/(2m), there is a gage δ on A such that

∑_{i=1}^{p}|f(x_{i})|A_{i}| − F (A_{i})| < ε

for each ε-regular δ-fine partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})} in A. Choose a gage Δ on N so that Δ ≤ δ ↾ N, N ∩∂A ⊂ Z_{Δ}, and B_{Δ(x)}(x) ⊂ A for each x ∈ N ∩int A. Now if Q = {(B_{1}, y_{1}), …, (B_{q}, y_{q})} is an ε-regular Δ-fine partition, then Q is a δ-fine partition in A. Hence

∑_{j=1}^{q}|F (B_{j})| = ∑_{j=1}^{q}|f(y_{j})|B_{j}| − F (B_{j})| < ε ,

and we conclude V_{∗}F (N) ≤ ε. The arbitrariness of ε implies V_{∗}F is AC.

Theorem 4.6 in conjunction with Corollary 3.4 provides an elegant comparison of the generalized Riemann and Lebesgue integrals.

Corollary 4.7. Let F be a charge in a figure A.

• If V_{∗}F is AC, then F is the indefinite R-integral of DF.

• If V_{∗}F is AC and finite, then F is the indefinite Lebesgue integral of DF.

5 Proof of Theorem 4.1

Throughout this section a number s with 0 < s < m, and an integer N with N^{m−s} > 2 remain fixed. For n = 0, 1, …, the product

∏_{i=1}^{m} [\frac{k_{i}}{N^{n}} , \frac{k_{i} + 1}{N^{n}}] ,

where k_{1}, …, k_{m} are integers, is called an n-cube. Denote by Q(n) the family of all n-cubes, and let Q = ⋃_{n=0}^{∞} Q(n). For i = 1, 2, …, select an integer

p_{i} ≥ \frac{(m − s) + is}{m − s}  and let  η_{i} = \frac{2N^{m(p_{i}+i)}}{2N^{m(p_{i}+i)} − 1}

A set C ⊂ ℝ^{m} is called amiable if it is compact and supporting, and if there are positive integers n_{i} such that for i = 1, 2, …, we have n_{i+1} > n_{i} + p_{i} + i and |C ∩ Q| ≥ |Q|/η_{i} whenever Q ∈ Q(n_{i}) meets C. We prove Theorem 4.1 by showing that each measurable set of positive measure has an amiable subset, and that amiable sets have the desired properties.

Lemma 5.1. Let K ⊂ R^{m} be a compact set with |K| > 0 and let k ≥ 0 be an integer. Suppose Q_{j} ∈ Q and ν_{j} > 1 are chosen so that |K ∩ Q_{j}| > |Q_{j}|/ν_{j} for j = 1, …, k. If ν > 1, then for each sufficiently large integer n ≥ 1 there is a nonempty compact set K_{n} ⊂ K such that |K_{n} ∩ Q_{j}| > |Q_{j}|/ν_{j} for j = 1, …, k, and |K_{n} ∩ Q| > |Q|/ν for each Q ∈ Q(n) which meets K_{n}.

Proof. If x ∈ K is a density point of K, there is an integer n_{x} ≥ 1 such that |K ∩ Q| > |Q|/ν for each Q ∈ ⋃_{n≥n_{x}} Q(n) containing x. Letting

E_{n} = {x ∈ K : n_{x} ≤ n} ,

the Lebesgue density theorem implies lim |E_{n}| = |K|, and hence lim |E_{n} ∩Q_{j}| = |K ∩ Q_{j}| for j = 1, …, k. Thus for each sufficiently large integer n ≥ 1 we have |E_{n}| > 0 and |E_{n} ∩ Q_{j}| > |Q_{j}|/ν_{j} for j = 1 …, k. Fix such an n, and let

K = {Q ∈ Q(n) : E_{n} ∩ Q ≠ ∅} and K_{n} = K ∩ ⋃ K .

As the family K is finite, K_{n} is compact, and since E_{n} ⊂ K_{n}, the set K_{n} is not empty and |K_{n} ∩ Q_{j}| ≥ |E_{n} ∩ Q_{j}| > |Q_{j}|/ν_{j} for j = 1, …, k. If Q ∈ Q(n) meets K_{n}, then Q ∈ K. Hence K_{n} ∩ Q = K ∩ Q and E_{n} ∩ Q ≠ ∅. In particular, Q contains an x ∈ K with n_{x} ≤ n, which implies |K_{n} ∩ Q| = |K ∩ Q| > |Q|/ν.

Corollary 5.2. Each measurable set E ⊂ ℝ^{m} with |E| > 0 has an amiable subset.

Proof. Select a compact set K ⊂ E with |K| > 0. Using Lemma 5.1, construct inductively positive integers n_{i} and nonempty compact sets K_{i} ⊂ K so that for i = 1, 2, …, we have

n_{i+1} > n_{i} + p_{i} + i , K_{i+1} ⊂ K_{i} ,

and |K_{i} ∩ Q| > |Q|/η_{k} whenever k ≤ i and Q ∈ Q(n_{k}) meets K_{i}. Having done this, let K_{0} = ⋂_{i=1}^{∞} K_{i}, and observe

|K_{0} ∩ Q| = lim_{i→∞} |K_{i} ∩ Q| ≥ \frac{|Q|}{η_{k}}

for each Q ∈ Q(n_{k}) which meets K_{0}. Now |K_{0}| > 0 since K_{0} ≠ ∅, and it suffices to let C = supp (λ L K_{0}).

To justify the inductive construction, first apply Lemma 5.1 to k = 0 and ν = η_{1}, and obtain an integer n_{1} ≥ 1 and a nonempty compact set K_{1} ⊂ K such that |K_{1} ∩ Q| > |Q|/η_{1} for each Q ∈ Q(n_{1}) which meets K_{1}. Next assume positive integers n_{2}, …, n_{i} and nonempty compact sets K_{2}, …, K_{i} have been constructed so that for j = 2, …, i, we have

n_{j} > n_{j−1} + p_{j−1} + (j − 1) , K_{j} ⊂ K_{j−1} ,

and |K_{j} ∩ Q| > |Q|/η_{k} whenever Q ∈ Q(n_{k}) meets K_{j} and 1 ≤ k ≤ j. Since only finitely many Q ∈ ⋃_{k=1}^{i} Q(n_{k}) meet K_{i}, Lemma 5.1 implies there is a an integer n_{i+1} > n_{i} + p_{i} + i and a nonempty compact set K_{i+1} ⊂ K_{i} such that |K_{i+1} ∩ Q| > |Q|/η_{k} whenever Q ∈ Q(n_{k}) meets K_{i+1} and 1 ≤ k ≤ i + 1.

Let C ⊂ ℝ^{m} be an amiable set, and let {n_{i}} be the associated sequence of positive integers. For each integer i ≥ 1 and each Q ∈ Q(n_{i} + p_{i}) select a Q^{∗} ∈ Q(n_{i} + p_{i} + i), and let

H_{i} = ⋃_{Q∈Q(n_{i}+p_{i})} int Q^{∗} and H_{C} = ⋂_{j=1}^{∞} ⋃_{i=j}^{∞} H_{i} .

Each ⋃_{i=j}^{∞} H_{i} is a dense open subset of ℝ^{m}, and so H_{C} is a dense G_{δ} subset of ℝ^{m} by the Baire category theorem. Every Q ∈ Q(0) contains N^{m(n_{i}+p_{i})} connected components of H_{i}, and each connected component of H_{i} has measure N^{−m(n_{i}+p_{i}+i)}. Thus |Q ∩ H_{i}| = N^{−mi} < 2^{−i} by our choice of N, and

|Q ∩ H_{C}| = lim_{j→∞} ∑_{i=j}^{∞} |H_{i}| = 0 .

We conclude H_{C} = ⋃_{Q∈Q(0)}(Q ∩ H_{C}) is a negligible set.

Observation 5.3. If Q ∈ Q(n_{i}) meets C, then |C ∩ Q^{∗}| ≥ |Q^{∗}|/2 for each connected component Q^{∗} of H_{i} ∩ Q.

Proof. As C is an amiable set,

|Q − C| = |Q| − |Q ∩ C| ≤ (1 − \frac{1}{η_{i}} ) |Q| = \frac{|Q|}{2N^{m(p_{i}+i)}} = \frac{|Q^{∗}|}{2} ,

and hence

|Q^{∗} ∩ C| = |Q^{∗}| − |Q^{∗} − C| ≥ |Q^{∗}| − |Q − C| ≥ \frac{|Q^{∗}|}{2} .

Corollary 5.4. The set H = C ∩ H_{C} is dense in C.

Proof. Let j ≥ 1 be an integer, and select a Q ∈ Q(n_{i}) with i ≥ j and Q∩C ≠ ∅. Then Q ∩ (C ∩ H_{i}) ≠ ∅ by Observation 5.3. We infer each Q ∈ ⋃_{i=j}^{∞} Q(n_{i}) that meets C meets also C ∩ ⋃_{i=j}^{∞} H_{i}. Since ⋃_{i=j}^{∞} Q(n_{i}) is a base for the topology of ℝ^{m}, the set C ∩ ⋃_{i=j}^{∞} H_{i} is a relatively open dense subset of C. The corollary follows from the Baire category theorem.

If A is a family of sets and B is a set, we let A_{B} = {A ∈ A : A ⊂ B}. For the purposes of this section only, we introduce the following terminology.

Let Z ⊂ ℝ^{m} and ε > 0. An ε-cover of Z is a nonoverlapping family Z ⊂ Q satisfying the following conditions:

1. Z ⊂ int ⋃ Z;

2. ∑_{B∈Z} d(B)^{s} < ε;

3. ∑_{B∈Z_{Q}} d(B)^{s} < d(Q)^{s} for each Q ∈ Q − Z.

Lemma 5.5. If Z ⊂ ℝ^{m} and H^{s}(Z) = 0, then Z has an ε-cover for each ε > 0.

Proof. The existence of a nonoverlapping family C(0) that satisfies conditions 1 and 2 of an ε-cover of Z follows from [5, Theorem 5.1]. Proceeding inductively, for n = 0, 1, …, construct families C(n) and D(n) so that D(n) consists of all D ∈ Q(n) − C(n) for which d(D)^{s} ≤ ∑_{B∈C(n)_{D}} d(B)^{s}, and

C(n + 1) = C(n) − ⋃_{D∈D(n)} C(n)_{D} .

Let C = ⋂_{n=1}^{∞} C(n) and D = ⋃_{n=1}^{∞} D(n), and observe that Z = C ∪ D is a nonoverlapping family with ⋃ C(0) ⊂ ⋃ Z; in particular, Z ⊂ int ⋃ Z. Furthermore,

∑_{B∈Z} d(B)^{s} = ∑_{B∈C} d(B)^{s} + ∑_{n=1}^{∞} ∑_{D∈D(n)} d(D)^{s}

≤ ∑_{B∈C} d(B)^{s} + ∑_{n=1}^{∞} ∑_{D∈D(n)} ∑_{B∈C(n)_{D}} d(B)^{s}

= ∑_{B∈C} d(B)^{s} + ∑_{B∈C(0)−C} d(B)^{s} = ∑_{B∈C(0)} d(B)^{s} < ε .

Select a Q ∈ Q(n) − Z, and observe that Q ∉ C(n). Indeed, if k ≥ n is an integer, then C(k) and C(n) contain the same n-cubes; thus Q ∈ C(n) implies Q ∈ C, a contradiction. Since Q ∉ D(n), we have

d(Q)^{s} > ∑_{B∈C(n)_{Q}} d(B)^{s} = ∑_{B∈C_{Q}} d(B)^{s} + ∑_{k=n}^{∞} ∑_{D∈D(k)_{Q}} ∑_{B∈C(k)_{D}} d(B)^{s}

≥ ∑_{B∈C_{Q}} d(B)^{s} + ∑_{k=n}^{∞} ∑_{D∈D(k)_{Q}} d(D)^{s}

= ∑_{B∈C_{Q}} d(B)^{s} + ∑_{D∈D_{Q}} d(D)^{s} = ∑_{B∈Z_{Q}} d(B)^{s} .

Observation 5.6. Let Q ∈ Q and let B ⊂ Q be a nonoverlapping family such that |Q ∩ ⋃ B| ≥ |Q|/2. Then d(Q)^{s} ≤ ∑_{B∈B} d(B)^{s}, and the equality occurs if and only if B = {Q}.

Proof. As the other cases are clear, assume B consists of proper subsets of Q, and observe that d(Q)/d(B) ≥ N for each B ∈ B. As N^{m−s} > 2, we obtain

d(Q)^{s} = \frac{|Q|}{d(Q)^{m−s}} ≤ \frac{2}{d(Q)^{m−s}} |⋃ B| = \frac{2}{d(Q)^{m−s}} ∑_{B∈B} |B|

≤ 2 [\frac{d(B)}{d(Q)}]^{m−s} ∑_{B∈B} d(B)^{s} ≤ \frac{2}{N^{m−s}} ∑_{B∈B} d(B)^{s} < ∑_{B∈B} d(B)^{s} .

Lemma 5.7. Let Q ∈ Q(n_{i}) and let B ⊂ Q be a nonoverlapping family of proper subcubes of Q such that |Q^{∗}∩⋃ B| ≥ |Q^{∗}|/2 for each connected component Q^{∗} of H_{i} ∩ Q. Then d(Q)^{s} < ∑_{B∈B} d(B)^{s}.

Proof. Enumerate Q(n_{i} + p_{i})_{Q} as {Q_{1}, …, Q_{r}} where r = N^{m(p_{i}+i)}, and let Q_{j}^{∗} = Q_{j} ∩ H_{i} for j = 1, …, r. We may assume that for an integer k with 0 ≤ k ≤ r, each Q_{1}, …, Q_{k} is contained in a B ∈ B, and no Q_{k+1}, …, Q_{r} is contained in any B ∈ B. Denote by C the family of all B ∈ B which contain one of the Q_{1}, …, Q_{k}, and observe that each B ∈ B − C is properly contained in one of the Q_{k+1}, …, Q_{r}. If k + 1 ≤ j ≤ r, then

|Q_{j}^{∗}| ≤ 2 |Q_{j}^{∗} ∩ ⋃ B| = |Q_{j}^{∗} ∩ ⋃ B_{Q_{j}} | ,

and Observation 5.6 implies d(Q_{j}^{∗})^{s} ≤ ∑_{B∈B_{Q_{j}}} d(B)^{s}. Now Q_{j}^{∗} is the interior

of an (n_{i} + p_{i} + i)-cube, and hence d(Q_{j}^{∗}) = N^{−(p_{i}+i)}d(Q) for j = 1, . . .. Since (m − s)p_{i} − si ≥ m − s by our choice of p_{i}, we obtain

∑_{B∈B−C} d(B)^{s} = ∑_{j=k+1}^{r} ∑_{B∈B_{Q_{j}}} d(B)^{s} ≥ ∑_{j=k+1}^{r} d(Q_{j}^{∗})^{s} = (r − k)d(Q)^{s}N^{−s(p_{i}+i)}

= (r − k)d(Q)^{s}N^{−mp_{i}}N^{(m−s)p_{i}−si} ≥ \frac{r − k}{r} d(Q)^{s}N^{m−s} .

On the other hand, ⋃_{j=1}^{k} Q_{j} ⊂ ⋃ C, |Q_{j}| = |Q|/r for j = 1, …, k, and d(B) ≤ d(Q)/N for each B ∈ B. Thus

\frac{k}{r} |Q| = ∑_{j=1}^{k} |Q_{j}| ≤ ∑_{B∈C} |B| = ∑_{B∈C} d(B)^{s}d(B)^{m−s}

≤ \frac{d(Q)^{m−s}}{N^{m−s}} ∑_{B∈C} d(B)^{s} = \frac{|Q|}{d(Q)^{s}N^{m−s}} ∑_{B∈C} d(B)^{s} ,

which yields ∑_{B∈C} d(B)^{s} ≥ (k/r)d(Q)^{s}N^{m−s}. We conclude

∑_{B∈B} d(B)^{s} = ∑_{B∈C} d(B)^{s} + ∑_{B∈B−C} d(B)^{s} ≥ d(Q)^{s}N^{m−s} > d(Q)^{s} .

Proof of Theorem 4.1. By Corollary 5.2, the set E has an amiable subset C. If {n_{i}} is the sequence of positive integers associated with C, define the set H_{C} as in the paragraph preceding Observation 5.3. Then |H_{C}| = 0, and H = C ∩ H_{C} is a G_{δ} subset of C, which is dense in C by Corollary 5.4. Choose a set Z ⊂ ℝ^{m} with H^{s}(Z) = 0, and a positive ε < |C|. In view of Lemma 5.5 there is an ε-cover Z of Z, and we let G = int ⋃ Z and C_{Z} = C − G.

Since Q is a base for the topology of ℝ^{m}, the set C_{Z} is supporting whenever |Q∩C_{Z}| > 0 for each Q ∈ Q that meets C_{Z}. Proceeding towards a contradiction, suppose there is a Q ∈ Q(n_{i}) with Q ∩ C_{Z} ≠ ∅ and |Q ∩ C_{Z}| = 0; in particular Q ∉ Z. As C is an amiable set, |Q ∩ C| ≥ |Q|/η_{i} > |Q|/2, and hence

|Q ∩ ⋃ Z_{Q}| = |Q ∩ ⋃ Z| ≥ |Q ∩ G| ≥ |Q ∩ C| > \frac{|Q|}{2} .

A contradiction d(Q)^{s} < ∑_{B∈Z_{Q}} d(B)^{s} follows from Observation 5.6.

It remains to show that H ∩ C_{Z} is a dense subset of C_{Z}. To this end, let j ≥ 1 be an integer, and select a Q ∈ Q(n_{i}) with i ≥ j and Q ∩ C_{Z} ≠ ∅; in particular Q ∉ Z. Proceeding towards a contradiction, assume Q ∩ (C_{Z} ∩ H_{i}) = ∅. Thus if Q^{∗} is a connected component of Q ∩ H_{i}, then Q^{∗} ∩ C ⊂ Q^{∗} ∩ G, and Observation 5.3 yields

|Q^{∗} ∩ ⋃ Z_{Q}| = |Q^{∗} ∩ ⋃ Z| ≥ |Q^{∗} ∩ G| ≥ |Q^{∗} ∩ C| ≥ \frac{|Q|}{2} .

A contradiction d(Q)^{s} < ∑_{B∈Z_{Q}} d(B)^{s} follows from Lemma 5.7. We conclude each Q ∈ ⋃_{i=j}^{∞} Q(n_{i}) that meets C_{Z} meets also C_{Z} ∩⋃_{i=j}^{∞} H_{i}. Since ⋃_{i=j}^{∞} Q(n_{i}) is a base for the topology of ℝ^{m}, the set C_{Z} ∩ ⋃_{i=j}^{∞} H_{i} is a relatively open dense subset of C_{Z}. The proposition follows from the Baire category theorem.

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