Lipeomorphisms, sets of bounded variation and integrals

Zoltán Buczolich

ABSTRACT

To obtain a lipeomorphic (bi-Lipschitz) change of variables theorem for generalized integrals lipeomorphic transformations of partitions are studied.

1. Introduction

For a sufficiently general setting of the Gauss-Green theorem in [P1] generalized integrals over sets of bounded variation were considered. The family of BV sets (sets of bounded variation, or sets of finite perimeter) provides the largest class of sets in R^{m} for which the surface area and exterior normal can still be defined in a way which leads to a useful integration theory. Originally in the Henstock type definition of higher dimensional integrals the integration was done on (higher dimensional) intervals and the partitions considered consisted of intervals as well. This concept was not coordinate free and one could not consider a sufficiently general class of transformations of these integrals. The current generalizations of the Henstock-Kurzweil integral are leading to interesting and difficult questions in geometric measure theory.

Instead of intervals one can consider the class of BV sets, or the class of figures (finite unions of intervals). In [BP] variations of additive functions were used to define two integrals. One is the F integral, based on partitions into figures. The other one is the BV integral, based on partitions into BV sets (the class BV consists of essentially closed BV sets, for each BV set A one can find a set A′ ∈ BV such that the symmetric difference of A and A′ is of measure zero). First, it is not clear how different these two concepts of integration are. In [P2] it was shown that if A is a figure then the class of F and BV integrable functions coincides. Therefore if we consider functions defined on figures we will only say that a function is integrable, since this will mean both F and BV integrability. On the other hand, a remark of [P2] shows that the BV integral is a proper generalization of the F integral. In fact, an example of this author was used to show that there is a BV set A, and a function f : A → R which has no integrable extension to a figure containing A.

Since in most cases the integral based on a smaller family of sets turns out to be more general this relationship of the BV and F integrals is a bit unusual.

Research supported by the Hungarian National Foundation for Scientific Research, Grant No. T 019476 and FKFP 0189/1997.

W. F. Pfeffer suggested to consider the following class of integrable functions. Given a BV set A, consider those functions f : A → R for which there is a figure K containing A such that f can be extended to be an integrable function on K. We will denote this class of functions by BVF(A). As the above mentioned example shows there are examples of BV sets when the class BVF(A) is a proper subset of BV(A). It would be interesting to see a characterization of those sets where these two concepts coincide. Clearly, figures are such sets.

One natural motivation for studying the BVF integral is that it is easy to see that it is additive, while for the BV integral an extension process, named the c-integral, is needed to obtain additivity, for details see [P1, Remark 5.2.5 and Section 9].

Working with BV sets the natural class of transformations is the class of lipeomorphisms. A transformation Φ : A → R^{m} is a lipeomorphism if both Φ and its inverse Φ^{−1} are Lipschitz mappings. Some authors call these mappings bi-Lipschitz mappings.

W. F. Pfeffer asked this author whether the BVF integral has a lipeomorphic change of variables property, that is, whether this class is invariant with respect to lipeomorphic transformations. This leads to the following, in our opinion very interesting, extension problem. Assume that A ⊆ R^{m} is a BV set and Φ : A → R^{m} is a lipeomorphism. Suppose furthermore, that K ⊆ R^{m} is a figure which contains Φ(A), and f : K → R is an integrable function (we have this situation if we consider BVF integrable functions on Φ(A)). Denote by J_{Φ}(x) the Jacobian of Φ where it exists. Recall that for lipeomorphisms J_{Φ} exists almost everywhere on A. Then, one can define g almost everywhere on A by setting g(x) = J_{Φ}(x)f(Φ(x)) at all points where the right hand side is defined. Then there is the natural question whether g(x) is BVF integrable on A, that is, whether there is an extension of g onto a figure containing A. In the main theorem of this paper we give a positive answer to this question. This also yields the lipeomorphic change of variables formula BVF ∫_{Φ(A)} fdλ = BVF ∫_{A}(f ∘ Φ)J_{Φ}dλ for the BVF integral.

This paper is organized in the following way. Studying generalized integrals, BV-sets, and lipeomorphisms one can see several interesting and difficult problems. After some preliminaries in the “Geometric Lemmas” section we review some of these properties. This author proved in [Bu1] that if Φ : A → R^{m} is a lipeomorphism then Φ maps density points of A exactly onto density points of Φ(A). It is easy to see that almost every density point is mapped onto density points, the main difficulty is showing that there is no exceptional set of measure zero. The basic problem is that Φ does not always have a lipeomorphic extension from A onto R^{m}. This will be the most difficult problem in this paper as well. To control the shape of the partitions only regular partitions are considered. This means that the product of the surface and the diameter of our sets cannot be too large compared to the volume of these sets. The main problem is that we will have to consider regular sets A_{i} such that we have no information about the regularity of A_{i} ∩ A. Since Φ does not necessarily extend to A_{i} as a lipeomorphism we face some difficulties when we need to do estimations related to the Φ image of A_{i}. In the “Main lemmas about lipeomorphic transformations of partitions” section we verify two Lemmas about this lipeomorphic changes of regular partitions. In Lemma 4.1 we deal with the case when there is enough room in the complement of Φ(A) in order to “make” the images Φ(A_{i} ∩ A) regular. In Lemma 4.2, which is the most delicate lemma of this paper, we deal with the case when there is not enough room in the complement of Φ(A). Finally in the last section we turn to generalized integrals, state and prove the main result of the paper.

2. Preliminaries

The set of reals is denoted by R. The Euclidean distance of x, y ∈ R^{m} is denoted by dist(x, y). The open ball centered at x ∈ R^{m} and of radius r is denoted by B(x, r). If A ⊆ R^{m} then B(A, r) = {x ∈ R^{m} : dist(x, A) < r}.

An interval in R^{m} is the Cartesian product of intervals on the coordinate axes. A figure is the finite union of closed compact non-degenerate intervals. The class of figures in R^{m} is denoted by F.

Given x ∈ R^{m} and d > 0 we denote by Q(x, d) the closed cube centered at x and of side length 2d. If a cube Q is given we denote by ℓ(Q) its side length, that is, ℓ(Q(x, d)) = 2d.

Definition 2.1. The set A is η-filling in the cube Q if from Q(x, η) ⊆ Q it follows that Q(x, η) ∩ A ≠ 0.

If A ⊆ R^{m} then d(A), cl A, int A, ∂A, and |A| will denote its diameter, closure, interior, boundary, and Lebesgue measure, respectively.

If A ⊆ R^{m} then the transformation Φ : A → R^{m} is Lipschitz when there is a constant c_{Φ} such that dist(Φ(x), Φ(y)) ≤ c_{Φ}dist(x, y) holds for all x, y ∈ A. We say that Φ is a c_{Φ}-lipeomorphism if both Φ and its inverse Φ^{−1} are c_{Φ}-Lipschitz transformations, that is,

dist(x, y)/c_{Φ} ≤ dist(Φ(x), Φ(y)) ≤ c_{Φ}dist(x, y),

observe that in this case c_{Φ} ≥ 1 should hold. If Φ : A → R^{m} is a c_{Φ}-Lipschitz mapping then by Kirszbraun’s theorem [Fe, 2.10.43] it has a c_{Φ}- Lipschitz extension Φ̃ : R^{m} → R^{m}. Unfortunately, there are no such extension theorems for c_{Φ}-lipeomorphisms. The best we can say, and the reader can easily check, that if Φ : A → R^{m} is a c_{Φ}- lipeomorphism then we can extend it to a c_{Φ}-lipeomorphism defined on cl A.

All sets considered in this paper will be Lebesgue measurable. The sets A, B ⊆ R^{m} are non-overlapping if |A∩B| = 0. The s-dimensional Hausdorff measure of the set A ⊆ R^{m} is denoted by H^{s}(A). For the special case s = m−1 we write simply H(A). A set is thin if it is of σ-finite H-measure. Also recall that there is a constant, κ_{m} such that H^{m}(A) = κ_{m}|A|.

Definition 2.2. Let 0 ≤ s < ∞, A ⊆ R^{m}, and a ∈ R^{m}. The upper s density of A at a is

Θ^{∗s}(A, a) = lim sup_{r→0+} \frac{H^{s}(A ∩ B(a, r))}{(2r)^{s}}.

If A ⊆ R^{m} we say that x is a dispersion point of A if Θ^{∗m}(A, x) = 0 and x is a density point of A if it is a dispersion point of R^{m} \ A. The set of the density points of A is called the essential interior of A and is denoted by int^{∗}A. The set R^{m} \ int^{∗}(R^{m} \ A) is called the essential closure of A, and is denoted by cl^{∗}A. Finally the essential boundary of A equals ∂^{∗}A = cl^{∗}A \ int^{∗}A. The set A is essentially closed or essentially open if A = cl^{∗}A, or A = int^{∗}A.

A bounded measurable set is a set of bounded variation (abbreviated as BV set) if it has a finite perimeter, that is, H(∂^{∗}A) = ||A|| is finite. (BV sets are usually defined otherwise by saying that A is a BV set if the distributional gradient of its characteristic function is a vector-valued measure in R^{m}, whose variation, the surface measure of A, is finite. However by [Fe, 2.10.6, and 4.5.11] this definition is equivalent to the one we use.) The class of essentially closed BV sets is denoted by BV. Observe that F ⊆ BV and if A ∈ F then A = cl A = cl^{∗}A, and ∂A = ∂^{∗}A.

If A ∈ BV then cl^{∗}A ∈ BV, and the symmetric difference of A and cl^{∗}A is of zero Lebesgue measure.

Given ϵ > 0 we say that a BV set A is ϵ-regular if d(A)||A|| > 0 and |A|/d(A)||A|| > ϵ. Observe that there are several slightly different concepts of regularity, but from the point of view of our integration theory these concepts are equivalent.

3. Geometric Lemmas

From [Bu2] we recall a theorem which says that if a set E is a subset of a BV set H, and H is of not too much surface then the Lipschitz image of E can also be wrapped into a figure of not too big surface.

Theorem 3.1. Assume that E ⊆ R^{m} is compact, Φ : E → R^{m} is a Lipschitz c_{Φ}- mapping, and H is a BV set such that E ⊆ int^{∗}H. Then there exists a figure F, and a constant c_{L} depending on c_{Φ} and m such that Φ(E) ⊆ int (F), ||F || ≤ c_{L}||H||, and |F | ≤ c_{L}|H|.

We remark that if E is a figure then Φ(E) is not necessarily a BV set, that is ||Φ(E)|| can be infinite, for the details see [Bu2]. In [Bu2] we stated the above theorem with the conclusion Φ(E) ⊆ F. However using a slightly bigger value of c_{L} we can enlarge the figure F to obtain Φ(E) ⊆ int (F).

We will also use from [Z, Theorem 5.4.3., p.230] the

Relative isoperimetric inequality. If H ⊆ R^{m} is a BV set then for each ball B(x, r) ⊆ R^{m} we have

min\{|B(x, r) ∩ H|, |B(x, r) \ H|\}^{\frac{m−1}{m}} ≤ c_{riso}H(B(x, r) ∩ ∂^{∗}H),

where the constant c_{riso} depends only on the dimension.

We say that Q ⊆ R^{m} is a dyadic netcube if there exist integers n, k_{1}, ..., k_{m} such that Q is the Cartesian product of intervals of the form [(k_{j} − 1)/2^{n}, k_{j}/2^{n}], j = 1, ..., m.

The next lemma is roughly stating that for regular BV sets we have a certain uniformity of the density properties, that is, we can find sufficiently large cubes which contain a lot of the set, but not too much of its surface.

Lemma 3.2. Assume that ϵ, ν > 0, B ⊆ R^{m} is an ϵ-regular BV set and x ∈ B is an arbitrary point. Then there exists a constant c_{ϵ,ν} < 1/4\sqrt{m} depending on ϵ, ν and m such that we can find a dyadic net cube Q = Q(x′, d_{Q}) ⊆ B(x, 4d(B)\sqrt{m}) satisfying

c_{ϵ,ν}d(B) < d_{Q} ≤ 2c_{ϵ,ν}d(B),
H(int (Q) ∩ ∂^{∗}B) ≤ ν||Q||,

and

|Q \ cl^{∗}B| < ν|Q|.

Proof. Choose a cube Q_{0} ⊃ B such that it is the union of 2^{m} non-overlapping dyadic net cubes of side length ℓ(Q_{0})/2, Q_{0} ⊆ B(x, 4d(B)\sqrt{m}) and ℓ_{0} = ℓ(Q_{0}) ≤ 4d(B). Since B is ϵ-regular |B| ≥ ϵ||B||d(B); hence

ℓ_{0}^{m} = |Q_{0}| ≥ |B| ≥ ϵ||B||ℓ_{0}/4, that is, 4ℓ_{0}^{m−1}/ϵ ≥ ||B||.

By the (standard) isoperimetric inequality there exists a constant c_{iso} depending only on m such that |B|^{(m−1)/m} ≤ c_{iso}||B||. Thus for any ϵ-regular BV set B we have

|B| ≥ ϵ||B||d(B) ≥ \frac{ϵ}{c_{iso}} |B|^{\frac{m−1}{m}} d(B),

(3.1)  |B|^{1/m} ≥ \frac{ϵ}{c_{iso}} d(B),

|B| ≥ \frac{ϵ^{m}}{c_{iso}^{m}} d(B)^{m} ≥ \frac{ϵ^{m}}{c_{iso}^{m}4^{m}} ℓ_{0}^{m} = βℓ_{0}^{m},

where β is a constant depending only on ϵ and m.

Choose k ∈ N and denote by Q_{1}, ..., Q_{2^{km}} the 2^{km} non-overlapping net subcubes of Q_{0} with side length ℓ_{0}/2^{k}. We assume that k is so large that ℓ_{0}/2^{k} < 1/4\sqrt{m}.

Denote by [x] the integer part of x ∈ R. Next we verify that there are at least [β2^{km−1}] many Q_{j}’s satisfying

|Q_{j} ∩ cl^{∗}B| ≥ β |Q_{j}|/2 = βℓ_{0}^{m}/2^{km+1}.

Denote by J the set of those j’s for which the above inequality is satisfied. If #J < β2^{km−1} then

βℓ_{0}^{m} ≤ |B| = |cl^{∗}B| = ∑_{j=1}^{2^{km}} |Q_{j} ∩ cl^{∗}B| =

∑_{j∉J} |Q_{j} ∩ cl^{∗}B| + ∑_{j∈J} |Q_{j} ∩ cl^{∗}B| < 2^{km} \frac{βℓ_{0}^{m}}{2^{km+1}} + β2^{km−1} \frac{ℓ_{0}^{m}}{2^{km}} = βℓ_{0}^{m}

which is a contradiction. Thus #J ≥ [β2^{km−1}].

Denote by J_{1} the set of those j’s in J for which H(int (Q_{j}) ∩ ∂^{∗}B) ≥ cν||Q_{j}|| = νc′_{m}ℓ(Q_{j})^{m−1}, where c and c′_{m} are constants to be determined later so that they will depend only on ν and m. Then

4ℓ_{0}^{m−1}/ϵ ≥ ||B|| ≥ ∑_{j∈J_{1}} H(int (Q_{j}) ∩ ∂^{∗}B) ≥ νc′_{m}(ℓ_{0}/2^{k})^{m−1} · #J_{1}.

Hence

#J_{1} ≤ \frac{4 · 2^{k(m−1)}}{ϵνc′_{m}} < #J

if k is sufficiently large (observe that the value of k is not depending on B, it depends only on ϵ, ν and m).

Assume Q = Q_{j} for a j ∈ J \J_{1}. If 0 < c < 1 then H(int (Q)∩∂^{∗}B) < cν||Q|| < ν||Q||. It is also clear that Q ⊆ Q_{0} ⊆ B(x, 4d(B)\sqrt{m}). Finally, using the version of the relative isoperimetric inequality which is valid for cubes there is a constant c_{risoQ} depending only on the dimension such that

min\{|Q ∩ B|, |Q \ B|\}^{\frac{m−1}{m}} ≤ c_{risoQ}H(int (Q) ∩ ∂^{∗}B) < c_{risoQ}cν||Q|| =

c_{risoQ}c′_{m}νℓ_{0}^{m−1}/2^{k(m−1)} < (\frac{β}{2} ℓ_{0}^{m}/2^{km})^{\frac{m−1}{m}} = (\frac{β}{2} |Q|)^{\frac{m−1}{m}}

where in the last inequality we assumed that c, and hence c′_{m}, are both chosen sufficiently small, clearly this smallness depends on ν, m and ϵ, where the dependence on ϵ is implicit, since β depends on ϵ.

Since j ∈ J implies |Q_{j} ∩ B| = |Q ∩ B| ≥ β|Q|/2 we infer

|Q \ B| < (c_{risoQ}c′_{m}ν)^{\frac{m}{m−1}} |Q| ≤ |Q|

whenever c_{risoQ}c′_{m}ν < 1, which we can assume.

Lemma 3.3. Assume B ⊆ R^{m} is a figure, Q ⊆ R^{m} is a cube of side length d, and |B ∩ Q| ≤ |Q|/2 = d^{m}/2. Then there exists a constant c_{F} depending only on m such that

H(B ∩ ∂Q) ≤ c_{F} H(∂B ∩ int (Q)).

Proof. Assume that S denotes one of the (m − 1)-dimensional sides of Q. Clearly, it is enough to find a constant c′_{F}, depending only on m, such that H(B ∩ S) ≤ c′_{F} H(∂B ∩ int (Q)). If x ∈ S denote by ℓ_{x} the line through x which is perpendicular to the hyperplane containing S. Put S_{1} = {x ∈ B ∩ S : ℓ_{x} ∩ ∂B ∩ int (Q) ≠ ∅}, S_{2} = (B ∩ S) \ S_{1}. Since projections are not increasing measure we have

(3.2)  H(S_{1}) ≤ H(∂B ∩ int (Q)).

If x ∈ S_{2} then ℓ_{x} ∩ ∂B ∩ int (Q) = ∅ and hence |B ∩ Q| ≥ H(S_{2})d. Using the version of the relative isoperimetric inequality which works for cubes there is a constant c depending only on m such that

|B ∩ Q|^{\frac{m−1}{m}} ≤ cH(∂B ∩ int (Q)),

where we also used the assumption |B ∩ Q| ≤ d^{m}/2, which, on the other hand, implies d^{m}/2 ≥ H(S_{2})d, that is, d ≥ (2H(S_{2}))^{1/(m−1)}. Hence

cH(∂B ∩ int (Q)) ≥ |B ∩ Q|^{\frac{m−1}{m}} ≥ (H(S_{2})d)^{\frac{m−1}{m}} ≥
(H(S_{2})(2H(S_{2}))^{\frac{1}{m−1}} )^{\frac{m−1}{m}} = 2^{1/m}H(S_{2}).

This and (3.2) implies the required inequality.

Next we recall some results of [V]. These results are related to degree theory.

Lemma 3.4. Assume that Φ : A → R^{m} is a c_{Φ}-lipeomorphism and Φ̃ : R^{m} → R^{m} denotes its extension which is obtained by Kirszbraun’s theorem. Then there exist constants η_{Φ} > 0, and c_{G} > 0 depending on c_{Φ} and m such that if Q is a cube of side length 2d and A is η_{Φ}d-filling in Q then Φ̃(Q) contains a cube of side length 2c_{G}d.

Proof. Since A is η_{Φ}d-filling in Q if η_{Φ} > 0 is sufficiently small then there are points of A sufficiently close to the center of Q. Therefore, we can assume that there is an x ∈ A ∩ Q such that B(x, d/2) ⊆ Q. Using the argument of 3.16 of [V] for a small η_{Φ} > 0 we obtain that B(Φ(x), d/4c_{Φ}) ⊆ Φ̃(B(x, d/2)) ⊆ Φ̃(Q) and hence we can choose c_{G} such that Q(Φ(x), c_{G}d) ⊆ B(Φ(x), d/4c_{Φ}) ⊆ Φ̃(Q).

It is not difficult to see that the following “rescaled” version of [V, Theorem 3.17] is also valid.

Theorem 3.5. Assume η, d > 0. Suppose that Ψ : A_{0} → R^{m} is a c_{Ψ}-lipeomorphism, where A_{0} ⊆ B(0, ηd), 0 ∈ A_{0}, and Ψ(0) = 0. Suppose also that 0 < 2\sqrt{m}/η^{2} < 1/16c_{Ψ}^{3}(c_{Ψ}+ 1) and that B(0, ηd) ⊆ B(A_{0}, 2\sqrt{m}d/η). Then B(0, ηd/2c_{Ψ}) ⊆ B(Ψ(A_{0}), c_{Ψ}2\sqrt{m}d/η).

We use the above theorem to obtain estimates of the size of holes in lipeomorphic preimages. We state the next lemma the way it is used later.

Lemma 3.6. Assume that A ⊆ R^{m} and Φ : A → R^{m} is a c_{Φ}-lipeomorphism. Choose η > \sqrt{m} such that 2\sqrt{m}/η^{2} < 1/16c_{Φ}^{3}(c_{Φ} + 1). Then for any d > 0 and x ∈ A if Φ(A) is d/η-filling in Q(Φ(x), ηd) then A is c_{Φ}2\sqrt{m}d/η-filling in Q(x, ηd/2c_{Φ}\sqrt{m}).

Proof. Without limiting generality we can assume that x = Φ(x) = 0. Denote the inverse of Φ by Ψ. Then Ψ is also a c_{Φ}-lipeomorphism with Ψ(0) = 0. Assume that y_{0} ∈ B(0, ηd) ⊆ Q(0, ηd). Observe that ηd > d > \sqrt{m}d/η. It is easy to see that we can choose y ∈ B(y_{0}, \sqrt{m}d/η) such that B(y, \sqrt{m}d/η) ⊆ B(0, ηd). Then Q(y, d/η) ⊆ B(y, \sqrt{m}d/η) ⊆ B(0, ηd) ⊆ Q(0, ηd). By our assumption ∅ ≠ Q(y, d/η) ∩ Φ(A) ⊆ B(y_{0}, 2\sqrt{m}d/η) ∩ Φ(A), that is, letting A_{0} = Φ(A) ∩ B(0, ηd) we have Q(y, d/η) ∩ A_{0} ≠ ∅, therefore B(0, ηd) ⊆ B(A_{0}, 2\sqrt{m}d/η). Theorem 3.5 is applicable and we infer that

B(0, ηd/2c_{Φ}) ⊆ B(Ψ(A_{0}), c_{Φ}2\sqrt{m}d/η) ⊆ B(A, c_{Φ}2\sqrt{m}d/η).

Thus choose y′ such that

Q(y′, c_{Φ}2\sqrt{m}d/η) ⊆ Q(0, ηd/2c_{Φ}\sqrt{m}) ⊆ B(0, ηd/2c_{Φ}).

Then B(0, ηd/2c_{Φ}) ⊆ B(A, c_{Φ}2\sqrt{m}d/η) implies that there exists x ∈ A ∩ B(y′, c_{Φ}2\sqrt{m}d/η) ⊆ A ∩ Q(y′, c_{Φ}2\sqrt{m}d/η). This proves the Lemma.

Finally we need estimates of BV -derivatives of lipeomorphisms.

Lemma 3.7. Assume that K ⊆ R^{m} is a figure, A ⊆ R^{m} is a BV -set, Φ : A → K is a c_{Φ}-lipeomorphism. Then the following statements hold.

For every ϵ_{0} > 0 if A_{0} ⊆ A is an ϵ_{0}-regular BV -set (or BV-set) then the image Φ(A_{0}) is a c_{1}ϵ_{0}-regular BV -set (or BV-set), (where c_{1} depends only on c_{Φ} and m).

There exists an exceptional set A^{2} of measure zero with the property that for every ϕ : A → [1, +∞) “weight” function, and for every ϵ_{0} > 0 there exists a function η_{ϵ_{0}} > 0 such that for all x ∈ A \ A^{2} and for any ϵ_{0}-regular BV-set A_{0} ⊆ B(x, η_{ϵ_{0}}(x)) ∩ A we have

ϕ(x)|J_{Φ}(x)|A_{0}| − |Φ(A_{0})| | < ϵ_{0}|A_{0}|.

Proof. Since Φ is a lipeomorphism we can estimate by using [Fe, 2.10.11] the Hausdorff and Lebesgue measure of Φ images and preimages and we can choose a constant c depending only on c_{Φ} and m such that

d(A_{0})/c ≤ d(Φ(A_{0})) ≤ cd(A_{0}),
|A_{0}|/c ≤ |Φ(A_{0})| ≤ c|A_{0}|,

and

||A_{0}||/c ≤ ||Φ(A_{0})|| ≤ c||A_{0}||,

where for the last inequality one needs the property that ∂^{∗}(Φ(A_{0})) = Φ(∂^{∗}(A_{0})). The above three inequalities easily imply the first statement of the lemma.

From the argument of [P1, Theorem 6.7] about the BV -differentiation of lipeomorphisms it is not difficult to see that the second statement of the Lemma holds. We only remark that the set A^{2} can be chosen to be the set of points in A where the BV -derivative of Φ does not exist.

4. Main Lemmas about lipeomorphic transformations of partitions

Definition 4.1. The collection P = {(x_{1}, A_{1}), ..., (x_{p}, A_{p})} is an ϵ-regular F, or BV-pre-partition if the sets A_{i} are non-overlapping figures, or BV sets and the BV -sets {x_{i}} ∪ A_{i} are ϵ-regular.

If, in addition, x_{i} ∈ A_{i} holds for all i = 1, ..., p then P is a partition.

If A ⊆ R^{m} is a given set and x_{i} ∈ A holds for i = 1, ..., p then P is anchored in A. Finally we say that the partition P is in the set B ⊆ R^{m} if ∪_{i=1}^{p}A_{i} ⊆ B.

Definition 4.2. Assume that c > 1 is given. We say that P = {(x_{1}, A_{1}), ..., (x_{p}, A_{p})} and P′ = {(x′_{1}, A′_{1}), ..., (x′_{p}, A′_{p})} are c-equivalent partitions (or pre-partitions) if they are both partitions (or pre-partitions),

|A_{i}|/c ≤ |A′_{i}| ≤ c|A_{i}|,
||A_{i}||/c ≤ ||A′_{i}|| ≤ c||A_{i}||,

and

d(A_{i} ∪ {x_{i}})/c ≤ d(A′_{i} ∪ {x′_{i}}) ≤ c · d(A_{i} ∪ {x_{i}})

holds for i = 1, ..., p.

Remark. Since each partition is also a prepartition we can also say that a partition is c-equivalent to a prepartition. It is also clear that there is a constant c_{1} (which depends only on c) such that if the (pre-)partition P in the above definition is ϵ-regular then P′ is c_{1}ϵ-regular.

Lemma 4.1. Assume that ϵ > 0, A ⊆ R^{m}, and Φ : cl (A) → R^{m} is a c_{Φ}- lipeomorphism and P_{1} = {(x_{1}, A_{1}), ..., (x_{p}, A_{p})} is an ϵ-regular F-partition anchored in cl (A). Assume furthermore that there exists η > 1 such that for each i ∈ {1, ..., p}, Φ(A) is not d(A_{i})/η-filling in Q(Φ(x_{i}), ηd(A_{i})), that is, for each i we can choose y_{i} such that Q(y_{i}, d(A_{i})/η) ∩ Φ(A) = ∅, and Q(y_{i}, d(A_{i})/η) ⊆ Q(Φ(x_{i}), ηd(A_{i})).

Then there exists a constant c_{η,ϵ} > 1 depending on η, ϵ, c_{Φ} and m such that we can choose an F-pre-partition P_{2} = {(Φ(x_{1}), B_{1}), ..., (Φ(x_{p}), B_{p})} which is c_{η,ϵ}-equivalent to P_{1} and satisfies B_{i} ∩ Φ(cl^{∗}A) = ∅ for i = 1, ..., p.

Proof. For each i choose a dyadic net cube B′_{i} = Q(y′_{i}, d_{i}) such that B′_{i} ⊂ int (Q(y′_{i}, 2d_{i})) ⊂ Q(y_{i}, d(A_{i})/η) and d(A_{i})/8η ≤ d_{i} < d(A_{i})/4η.

Next we verify that there exists a constant κ > 1 depending on η, ϵ, c_{Φ} and m such that letting J_{i} = {j : B′_{j} ∩ B′_{i} ≠ ∅} we have #J_{i} < κ and for j ∈ J_{i} we have 1/κ ≤ d(B′_{j})/d(B′_{i}) ≤ κ. We can also assume that κ is an integer. Using B′_{j} ⊆ Q(y_{j}, d(A_{j})/η) ⊆ Q(Φ(x_{j}), ηd(A_{j})), from B′_{j} ∩ B′_{i} ≠ ∅ it follows that

(4.1)  Q(Φ(x_{j}), ηd(A_{j})) ∩ B′_{i} ≠ ∅.

Since Φ(x_{j}) ∈ Φ(A) and Φ(A) is disjoint from Q(y′_{i}, 2d_{i}) ⊃ B′_{i} = Q(y′_{i}, d_{i}), using (4.1) we infer that ηd(A_{j}) ≥ d_{i}. Thus d_{j} ≥ d(A_{j})/8η ≥ d_{i}/8η^{2}; hence d_{j}/d_{i} = d(B′_{j})/d(B′_{i}) ≥ 1/8η^{2}. By a similar, symmetrical argument we also have d_{i}/d_{j} ≥ 1/8η^{2}. Thus

(4.2)  8η^{2} ≥ d_{j}/d_{i} = d(B′_{j})/d(B′_{i}) ≥ 1/8η^{2}.

Since d(A_{i})/8η ≤ d_{i} < d(A_{i})/4η, and d(A_{j})/8η ≤ d_{j} < d(A_{j})/4η by (4.2) we obtain

d(A_{i})/16η^{2} ≤ d(A_{j}) ≤ 16η^{2}d(A_{i}).

Since {(x_{1}, A_{1}), ..., (x_{p}, A_{p})} is an ϵ-regular partition using an estimate like the one at (3.1) by the above inequality we can choose a constant c′(ϵ) > 0 depending on ϵ and m such that |A_{j}| ≥ c′(ϵ)d(A_{j})^{m} and by the above inequalities |A_{j}| ≥ c(ϵ)d(A_{i})^{m} with a constant c(ϵ) depending also on ϵ and m. Since B′_{i} ∩ B′_{j} ≠ ∅ implies Q(Φ(x_{i}), ηd(A_{i})) ∩ Q(Φ(x_{j}), ηd(A_{j})) ≠ ∅ we have

dist(Φ(x_{i}), Φ(x_{j})) ≤ η(d(A_{i}) + d(A_{j})) ≤ η(d(A_{i}) + 16η^{2}d(A_{i})) = d(A_{i})η(1 + 16η^{2}).

Since Φ is a c_{Φ}-lipeomorphism we have

dist(x_{i}, x_{j}) ≤ c_{Φ}d(A_{i})η(1 + 16η^{2}).

Thus

(4.3)  A_{j} ⊆ B(x_{i}, dist(x_{i}, x_{j}) + d(A_{j})) ⊆ B(x_{i}, c_{Φ}d(A_{i})η(1 + 16η^{2}) + 16η^{2}d(A_{i})).

Since the A_{j}’s are non-overlapping and |A_{j}| ≥ c(ϵ)d(A_{i})^{m} using (4.3) we infer that #J_{i} is bounded by a constant κ depending on η, ϵ, c_{Φ} and m. We can also assume that κ > 8η^{2} and hence κ ≥ d(B′_{j})/d(B′_{i}) ≥ 1/κ holds for each i, j. Since B′_{i} and B′_{j} are net cubes if they overlap either B′_{i} ⊆ B′_{j}, or B′_{j} ⊆ B′_{i}. Given i choose a j(i) (which might equal i) such that B′_{j(i)} ⊆ B′_{i} and there is no proper subinterval B′_{k} ⊆ B′_{j(i)}. Since we have only finitely many B′_{k}’s this minimal B′_{j(i)} should exist. Observe that B′_{j} ∩ B′_{i} ≠ ∅ and hence the number of i’s for which a given j_{0} = j(i) is bounded by κ.

It is also clear that d(B′_{j(i)}) ≥ d(B′_{i})/κ. Split B′_{j(i)} into κ^{m} many equal non-overlapping subcubes and choose one of these as B_{i} in a way that this cube has not been used for any i′ ≤ i. Using d(A_{i})/8η ≤ d_{i} < d(A_{i})/4η and the ϵ-regularity of P_{1} it is not difficult to see that there is a constant c_{η,ϵ}, depending on η, ϵ, c_{Φ} and m, such that {(Φ(x_{1}), B_{1}), ..., (Φ(x_{p}), B_{p})} is c_{η,ϵ}- equivalent to P_{1}. From B_{i} ⊆ B′_{i} ⊆ int (Q(y′_{i}, 2d_{i})) ⊆ int (Q(y_{i}, d(A_{i})/η)) it follows that B_{i} ∩ Φ(cl^{∗}A) = ∅ for each i.

Lemma 4.2. Assume that A ⊆ R^{m} is a BV -set and Φ : cl (A) → R^{m} is a c_{Φ}- lipeomorphism. Given ϵ > 0 there exist η_{0} > 2 and a constant c_{η_{0}} > 0 (both depending on ϵ, c_{Φ} and m) which satisfy the following property. If P_{1} = {(x_{1}, A_{1}), ..., (x_{p}, A_{p})} is an ϵ-regular F- partiton anchored in cl (A) and Φ(A) is d(A_{i})/η_{0}-filling in Q(Φ(x_{i}), η_{0}d(A_{i})) for each i, then we can choose an F-pre-partition P_{2} = {(Φ(x_{1}), B_{1}), ..., (Φ(x_{p}), B_{p})} which is c_{η_{0}}- equivalent to P_{1} and Φ(A_{j} ∩ cl^{∗}(A)) ∩ B_{i} = ∅ for i ≠ j and i, j ∈ {1, ..., p}.

Proof. The A_{i}’s are figures; hence they are closed and their essential closures, boundaries, and interiors coincide with their regular closures, boundaries, and interiors, respectively.

Since each A_{i} is ϵ-regular, with a constant 1/3 > ν > 0 use Lemma 3.2 to choose a dyadic net cube Q′_{i} = Q(x′_{i}, d′_{i}) ⊆ B(x_{i}, 4d(A_{i})\sqrt{m}), satisfying

(4.4)  c_{ϵ,ν}d(A_{i}) < d′_{i} ≤ 2c_{ϵ,ν}d(A_{i}) ≤ d(A_{i})/2\sqrt{m},
H(int (Q′_{i}) ∩ ∂A_{i}) ≤ ν||Q′_{i}||,

and

|Q′_{i} \ A_{i}| < ν|Q′_{i}|.

Assume that c′ > 0 is a small constant (it can be chosen to be 1/30c_{Φ}^{2}\sqrt{m}).

Set c′′_{G} = (c_{G}c′)^{m} where c_{G} is the constant from Lemma 3.4.

We can assume that ν is so small that c_{L}6ν < c′′_{G}/2 where c_{L} is the constant from Theorem 3.1. Since c_{L}, c′′_{G} depend on c_{Φ} and m our choice of ν will also depend only on c_{Φ} and m.

Assume that η_{0} > 2 + \sqrt{m} is fixed and sufficiently big, that is, it satisfies the inequalities given later in this proof. Since all these inequalities will give lower bounds for η_{0} they can all simultaneously be satisfied. These bounds will make η_{0} depend on ϵ, c_{Φ} and m.

First we assume that 2\sqrt{m}/η_{0}^{2} < 1/16c_{Φ}^{3}(c_{Φ}+1) and observe that, by Lemma 3.6, from Φ(A) is d(A_{i})/η_{0}-filling in Q(Φ(x_{i}), η_{0}d(A_{i})) it follows that A is c_{Φ}2\sqrt{m}d(A_{i})/η_{0}-filling in Q(x_{i}, η_{0}d(A_{i})/2c_{Φ}\sqrt{m}).

We can assume that η_{0} was chosen so large that c_{Φ}2\sqrt{m}d(A_{i})/η_{0} < d′_{i}/3 and B(x_{i}, 4d(A_{i})\sqrt{m}) ⊆ Q(x_{i}, η_{0}d(A_{i})/2c_{Φ}\sqrt{m}). Then Q′_{i} ⊆ Q(x_{i}, η_{0}d(A_{i})/2c_{Φ}\sqrt{m}) and there exists x′′_{i} ∈ Q(x′_{i}, d′_{i}/3) ∩ A. Put Q′′_{i} = Q(x′′_{i}, c′d′_{i}).

Next we can also assume that η_{0} is so large that using η_{Φ} from Lemma 3.4 we have

c_{Φ}2\sqrt{m}d(A_{i})/η_{0} < η_{Φ}c′d′_{i}.

Then Lemma 3.4 implies that Φ̃(Q′′_{i}) contains a cube of side length 2c_{G}c′d′_{i}. Denote this cube by B′′_{i}. Set c′_{G} = (c_{G}c′)^{m−1} and recall that c′′_{G} = (c_{G}c′)^{m}. Then ||B′′_{i}|| = c′_{G}||Q′_{i}|| ≤ c′_{G}||A_{i}|| and |B′′_{i}| = c′′_{G}|Q′_{i}|. Unfortunately, the cubes B′′_{i} cannot be used to define the desired pre-partition. The problem is that Q′′_{i} ⊆ Q′_{i} might contain some smaller cubes of the form Q′′_{j} ⊆ Q′_{j} and this implies that the sets B′′_{i} and B′′_{j} might overlap. Recall that the cubes Q′_{i} are dyadic netcubes and hence if Q′_{i} and Q′_{j} overlap then either Q′_{i} ⊆ Q′_{j}, or Q′_{j} ⊆ Q′_{i}. We have |Q′_{i} \ A_{i}| < ν|Q′_{i}| < |Q′_{i}|/3, and |Q′_{j} \ A_{j}| < ν|Q′_{j}| < |Q′_{j}|/3, thus i ≠ j, and Q′_{i} = Q′_{j} would imply that the non-overlapping figures A_{i} and A_{j} would both contain more than two-thirds of Q′_{i} = Q′_{j}, which is impossible; hence if Q′_{i} and Q′_{j} overlap then either Q′_{i} ⊂ Q′_{j}, or Q′_{j} ⊂ Q′_{i}. Given i denote by J_{i} the set of those indices j for which j ≠ i and Q′_{j} ⊂ Q′_{i}. Since we work with netcubes we can choose a suitable subset J_{i}^{∗} ⊆ J_{i} such that for j, j′ ∈ J_{i}^{∗}, j ≠ j′ the cubes Q′_{j} and Q′_{j′} are non-overlapping and ∪_{j∈J_{i}^{∗}}Q′_{j} = ∪_{j∈J_{i}}Q′_{j}.

Let the figure A′_{i} = cl ((Q′_{i} ∩ A_{i}) \ ∪_{j∈J_{i}}Q′_{j}) = cl ((Q′_{i} ∩ A_{i}) \ ∪_{j∈J_{i}^{∗}}Q′_{j}).

Since the figures A_{i} were non-overlapping we have A′_{i} ∩ int (A_{j}) = ∅ for j ∈ J_{i}. Furthermore, |Q′_{j} ∩ A_{i}| ≤ |Q′_{j} \ A_{j}| < ν|Q′_{j}| < |Q′_{j}|/3 for j ∈ J_{i}. From Lemma 3.3 we infer that H(A_{i} ∩ ∂Q′_{j}) ≤ c_{F} H(∂A_{i} ∩ int (Q′_{j})). By using (4.4) and the fact that the cubes Q′_{j}, j ∈ J_{i}^{∗} are non-overlapping we obtain

||A′_{i}|| ≤ ||A_{i}|| + ||Q′_{i}|| + ∑_{j∈J_{i}^{∗}} H(A_{i} ∩ ∂Q′_{j}) ≤ ||A_{i}|| + ||Q′_{i}|| + c_{F} ||A_{i}|| ≤ c′_{F} ||A_{i}||

with a constant c′_{F} depending only on c_{F} which depends only on m.

From |Q′_{j}| = |Q′_{j}∩A_{j}|+|Q′_{j}\A_{j}| < |Q′_{j}∩A_{j}|+|Q′_{j}|/3 it follows that |Q′_{j}| < 3|Q′_{j}∩A_{j}|/2. On the other hand, the figures A_{j} are non-overlapping and Q′_{j} ⊂ Q′_{i} for j ∈ J_{i}; hence

∑_{j∈J_{i}} |Q′_{j} ∩ A_{j}| = |Q′_{i} ∩ ∪_{j∈J_{i}}(Q′_{j} ∩ A_{j})| ≤ |Q′_{i} \ A_{i}| < ν|Q′_{i}|.

Thus |∪_{j∈J_{i}}Q′_{j}| ≤ ∑_{j∈J_{i}} |Q′_{j}| < (3/2) ∑_{j∈J_{i}} |Q′_{j}∩A_{j}| < (3ν/2)|Q′_{i}|. Since |Q′_{i}\A_{i}| < ν|Q′_{i}| we obtain |cl (Q′_{i} \ A′_{i})| = |Q′_{i} \ A′_{i}| < |Q′_{i} \ A_{i}| + ∑_{j∈J_{i}} |Q′_{j}| < 3ν|Q′_{i}|.

Let E_{i} = (Q′_{i} ∩ ∪_{j≠i}A_{j}) ∪ ∪_{j∈J_{i}}Q′_{j} and H′_{i} = cl (Q′_{i} \ A′_{i}) ⊃ E_{i}. Since Q′_{i} and A′_{i} are figures we can enlarge the figure H′_{i} a bit to obtain H_{i} such that H′_{i} ⊆ int (H_{i}), ||H_{i}|| < 2||H′_{i}|| and |H_{i}| < 2|H′_{i}|. Applying Theorem 3.1 find a figure F_{i} such that Φ̃(E_{i}) ⊆ int (F_{i}) and ||F_{i}|| ≤ c_{L}||H_{i}|| < 2c_{L}||H′_{i}|| ≤ 2c_{L}(||Q′_{i}|| + ||A′_{i}||) ≤ (2c′_{F} c_{L} + 1)||A_{i}||, and |F_{i}| ≤ c_{L}|H_{i}| ≤ 2c_{L}|H′_{i}| ≤ c_{L}6ν|Q′_{i}|.

Put B_{i} = cl (B′′_{i} \ F_{i}). Then

||B_{i}|| ≤ ||B′′_{i}|| + ||F_{i}|| ≤ (c′_{G} + 2c′_{F} c_{L} + 1)||A_{i}||

and our assumption that c_{L}6ν < c′′_{G}/2 and (4.4) imply

|B_{i}| ≥ |B′′_{i}| \ |F_{i}| ≥ c′′_{G}|Q_{i}| − c_{L}6ν|Q_{i}| ≥ c′′_{G}|Q′_{i}|/2 ≥ c̃|A_{i}|,

where c̃ depends only on c_{Φ} and m. Recalling that η_{0} depends on ϵ, c_{Φ} and m we can choose a constant c_{D} depending also on ϵ, c_{Φ}, and m such that

B_{i} ⊆ B′′_{i} ⊆ Φ̃(Q′′_{i}) ⊆ Φ̃(Q′_{i}) ⊆ Φ̃(Q(x_{i}, η_{0}d(A_{i})/2c_{Φ}\sqrt{m})) ⊆ B(Φ(x_{i}), c_{D}d(A_{i})).

The above three displayed formulae yield that there exists a constant c_{η_{0}} depending on ϵ, c_{Φ} and m such that P_{1} and P_{2} are c_{η_{0}}-equivalent.

Next we verify that the sets {B_{i}} are non-overlapping.

Thus, proceeding towards a contradiction, assume that B_{i} and B_{j} overlap. Then the corresponding dyadic net cubes are Q′_{i} = Q(x′_{i}, d′_{i}), and Q′_{j} = Q(x′_{j}, d′_{j}). Without limiting generality we can assume that d′_{i} ≥ d′_{j}.

If Q′_{i} and Q′_{j} overlap then Q′_{j} ⊂ Q′_{i} and j ∈ J_{i}. Then B_{j} ⊆ Φ̃(Q′′_{j}) ⊆ Φ̃(Q′_{j}) ⊆ Φ̃(E_{i}) ⊆ int (F_{i}) and hence B_{j} ∩ B_{i} = ∅ which is impossible.

Therefore, we can assume that Q′_{i} and Q′_{j} do not overlap.

Suppose that z ∈ cl (A) \ Q′_{i}. From x′′_{i} ∈ Q(x′_{i}, d′_{i}/3) ⊂ Q(x′_{i}, d′_{i}) = Q′_{i} it follows that dist(z, x′′_{i}) ≥ d′_{i}/3. Since Φ is a c_{Φ}-lipeomorphism this implies that dist(Φ(z), Φ(x′_{i})) ≥ d′_{i}/3c_{Φ}. On the other hand Φ̃(Q(z, c′d′_{i})) ⊆ B(Φ(z), \sqrt{m}c_{Φ}c′d′_{i}). Since c′ is sufficiently small we infer that Φ̃(Q(z, c′d′_{i})) ∩ Φ̃(Q′′_{i}) = Φ̃(Q(z, c′d′_{i})) ∩ Φ̃(Q(x′′_{i}, c′d′_{i})) = ∅. Since Q′_{i} and Q′_{j} do not overlap choosing z = x′′_{j} we have x′′_{j} ∈ cl (A)\Q′_{i} and by the above argument B_{i} ∩B_{j} ⊆ B′′_{i} ∩B′′_{j} ⊆ Φ̃(Q′′_{i})∩Φ̃(Q′′_{j}) ⊆ Φ̃(Q′′_{i})∩Φ̃(Q(x′′_{j}, c′d′_{j})) ⊆ Φ̃(Q′′_{i})∩Φ̃(Q(x′′_{j}, c′d′_{i})) = ∅.

We verified above that if z ∈ cl (A) \ Q′_{i} then Φ(z) ∉ B_{i}. Therefore Φ((A_{j} ∩ cl^{∗}A) \ Q′_{i}) ∩ B_{i} = ∅. On the other hand, Φ̃(Q′_{i} ∩ A_{j}) ⊆ Φ̃(E_{i}) ⊆ int (F_{i}) for j ≠ i and hence B_{i} ∩ Φ̃(Q′_{i} ∩ A_{j}) = ∅. Thus for j ≠ i

B_{i} ∩ Φ(A_{j} ∩ cl^{∗}(A)) ⊆ B_{i} ∩ (Φ((A_{j} ∩ cl^{∗}A) \ Q′_{i}) ∪ Φ̃(Q′_{i} ∩ A_{j})) = ∅.

This completes the proof of the lemma.

5. The main Theorem

To avoid repeating the same definition twice, first for figures and then for BV-sets in the sequel A stands for either F, or for BV.

Definition 5.1. A charge is a real valued function F which satisfies the next two conditions,

i) it is additive, that is, F (A ∪ B) = F (A) + F (B) when A, B ∈ A are nonoverlapping sets.

ii) it is continuous, that is for any ϵ > 0 there is an η > 0 such that |F (A)| < ϵ for each A ∈ A with A ⊆ B(0, 1/ϵ), ||A|| < 1/ϵ and |A| < η.

Definition 5.2. A non-negative real-valued function δ defined on the set A ⊆ R^{m} is called a gage function on A if its null set {x ∈ A : δ(x) = 0} is thin, that is, of σ-finite H-measure.

Definition 5.3. If A ∈ A we say that the real-valued function f defined on A is A-integrable if there is a charge F such that for any ϵ > 0 there is a gage δ on A satisfying

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

for each ϵ-regular δ-fine A-partition {(x_{1}, A_{1}), ..., (x_{p}, A_{p})} in A.

Remarks. By [P2] if A is a figure then the class of BV and F integrable functions is the same. Thus, working with figures sometimes we just simply say that a function is integrable, meaning that it is both F and BV integrable.

The integrability and the value of the integral of a function is not changed if we modify its definition on a set of measure zero.

Observe that by a slight modification BV-pre-partitions can be converted into BVpartitions. Since the charge in the definition of the integral is continuous one can easily see that if f is integrable on the figure K and ϵ > 0 is given then we can choose a gage function δ such that

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

holds for any ϵ-regular δ-fine BV-pre-partition {(x_{1}, A_{1}), ..., (x_{p}, A_{p})}. Indeed, for ϵ/2 > 0 choose a gage function δ such that

∑_{i=1}^{p} |f(x_{i})|A′_{i}| − F (A′_{i})| < ϵ/2

holds for any ϵ/2-regular δ-fine BV-partition {(x_{1}, A′_{1}), ..., (x_{p}, A′_{p})} anchored in K.

Using the continuity of F it is not difficult to find for an ϵ-regular δ-fine BV-pre-partition {(x_{1}, A_{1}), ..., (x_{p}, A_{p})} an ϵ/2-regular δ- fine BV-partition {(x_{1}, A′_{1}), ..., (x_{p}, A′_{p})} such that

∑_{i=1}^{p} ||f(x_{i})|A_{i}| − F (A_{i})| − |f(x_{i})|A′_{i}| − F (A′_{i})|| < ϵ/2.

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

Definition 5.4. If A ⊆ R^{m} is a BV -set and f : A → R^{m} is a given function then we say that f is BVF integrable on A if there exists an integrable function g on a figure A_{0} such that A ⊆ A_{0}, f = g almost everywhere on A and g = 0 outside A. Then BVF ∫_{A} f = F ∫_{A_{0}} g.

Theorem 5.1. Assume that K ⊆ R^{m} is a given figure, f : K → R is integrable, A ⊆ R^{m} is a BV set and Φ : A → K is a lipeomorphism. Then (f ∘ Φ)J_{Φ} is BVF integrable on A and BVF ∫_{A}(f ∘ Φ)J_{Φ} = BVF ∫_{Φ(A)} f.

Remark. The above theorem implies that if A is a BV -set, Φ : A → R^{m} is a lipeomorphism and f is BVF integrable on Φ(A) then (f ∘ Φ)J_{Φ} is BVF integrable on A.

Proof.

Step 1. Some definitions and assumptions.

Since changing our functions on sets of measure 0 does not change integrability we can assume that A ∈ BV, that is, cl^{∗}(A) = A. If x ∈ A and J_{Φ}(x) is defined put g(x) = f(Φ(x))J_{Φ}(x), otherwise set g(x) = 0.

Assume that A_{0} is a figure containing cl (A) in its interior. We need to verify that g is F-integrable on A_{0}. This clearly implies that g is integrable on any subfigure of R^{m}.

Assume that 0 < ϵ < 1 is given and fixed.

We can assume that Φ is extended onto the closure of A, that is, Φ is a cl (A) → K lipeomorphism with constant c_{Φ}.

Since Φ is a lipeomorphism density points are mapped onto density points, that is, Φ(cl^{∗}(A)) = cl^{∗}(Φ(A)) and we also have Φ(∂^{∗}A) = ∂^{∗}(Φ(A)).

Using that F ∫_{K} f = BV ∫_{K} f exists choose an additive continuous function F satisfying the definition of the integral. By the remark after the definition of pre-partitions for ϵ_{1} > 0 choose a gage function δ ≥ 0 such that

(5.1)  ∑_{i=1}^{p} |f(x_{i})|B_{i}| − F (B_{i})| < ϵ_{1}

for each ϵ_{1}-regular δ-fine BV-pre-partition {(x_{1}, B_{1}), ..., (x_{p}, B_{p})} in K. Here ϵ_{1} is a small constant which will depend on ϵ, c_{Φ} and m.

Since ∂K = ∂^{∗}K is thin we can also assume that δ(x) = 0 for x ∈ ∂K and for x ∈ int (K) we have B(x, δ(x)) ⊆ int (K).

Step 2. The definition of the gage function δ_{1} on A_{0}. This function will be used to show the integrability of g.

If x ∉ cl (A) choose δ_{1}(x) > 0 such that B(x, δ_{1}(x)) ∩ cl (A) = ∅.

It is a corollary of Theorem 6.2 of [M, p.89] that if A′ ⊆ R^{m} with H(A′) = H^{m−1}(A′) < ∞ then the upper (m − 1)-density Θ^{∗m−1}(A′, x) = 0 for H-almost every x ∉ A′. Using this theorem with A′ = ∂^{∗}A we obtain

lim sup_{r→0+} \frac{H(∂^{∗}A ∩ B(x, r))}{r^{m−1}} = 0 for H-almost every x ∉ ∂^{∗}A.

Denote by A^{1} the set of those x ∈ cl (A) \ ∂^{∗}A for which the above displayed equality does not hold. Then H(A^{1}) = 0, and put δ_{1}(x) = 0 for x ∈ A^{1}.

For x ∈ ∂^{∗}A also put δ_{1}(x) = 0, again recall that ∂^{∗}A is thin.

For x ∈ cl (A) \ (∂^{∗}A ∪ A^{1}) choose 0 < δ_{2}(x) such that
(5.2)  H(∂^{∗}A ∩ B(x, r)) < ϵ^{m−1}r^{m−1} holds for 0 < r < δ_{2}(x).
For x ∈ cl (A) \ (cl^{∗}A ∪ A^{1}) choose 0 < δ_{3}(x) < δ_{2}(x) such that
(5.3)  |cl^{∗}A ∩ B(x, r)| < \frac{ϵ_{1}r^{m}}{|f(Φ(x))| + 1} holds for 0 < r < δ_{3}(x).

For x ∈ int^{∗}(A) \ A^{1} choose 0 < δ_{3}(x) < δ_{2}(x) such that
(5.4)  |B(x, r) \ cl^{∗}(A)| < \frac{ϵ_{1}r^{m}}{(|f(Φ(x))| + 1)(J_{Φ}^{∗}(x) + 1)} holds for 0 < r < δ_{3}(x)

where J_{Φ}^{∗}(x) = J_{Φ}(x) if the latter is defined, otherwise J_{Φ}^{∗}(x) = 0.

Assume that 0 < c_{5} < 1, this constant will be determined later in a way that it will depend only on c_{Φ} and m. Using Lemma 3.7 choose a set of measure zero with the property that for ϕ(x) = |f(Φ(x))| + 1, ϵ_{0} = c_{5}ϵ, and x ∈ int^{∗}(A) \ A^{2} we can choose a δ_{4}(x) ≤ δ_{3}(x) such that

(5.5)  (|f(Φ(x))| + 1)|J_{Φ}(x)|B_{0}| − |Φ(B_{0})| | < c_{5}ϵ|B_{0}|

for any c_{5}ϵ-regular BV set B_{0} ⊆ B(x, δ_{4}(x)) ∩ A.

If x ∈ cl (A) \ (A^{1} ∪ ∂^{∗}A) and x ∉ int^{∗}(A) \ A^{2} then set δ_{4}(x) = δ_{3}(x).

Finally, for x ∈ cl (A) \ (A^{1} ∪ ∂^{∗}A) choose δ_{1}(x) > 0 such that δ_{1}(x) ≤ min{δ_{4}(x), c_{2}δ(Φ(x))}, where c_{2} ∈ (0, 1/c_{Φ}) is a small constant, to be determined later.

We remind that δ was chosen to obtain (5.1). Since Φ is a lipeomorphism Φ^{−1}({x : δ(x) = 0} ∩ cl (A)) is a thin set; hence {x : δ_{1}(x) = 0} is also thin.

Step 3. The defintion of F_{0}.

For B_{0} ∈ BV put F_{0}(B_{0}) = F (Φ(B_{0} ∩ cl^{∗}A)) where we define F (∅) = 0.

From H(∂^{∗}A) < ∞ it follows that ||B_{0} ∩ cl^{∗}A|| ≤ ||B_{0}|| + ||cl^{∗}A|| and hence B_{0} ∩ cl^{∗}A is a BV -set. By Lemma 3.7, Φ(B_{0} ∩ cl^{∗}(A)) is also a BV -set. Since Φ is a lipeomorphism using the additivity and continuity of F it is not difficult to see that F_{0} is also a charge.

Step 4. First step towards the estimation of the Riemannian sums of g, the index sets I_{j}.

Recall that A_{0} is a large figure such that cl (A) ⊆ int (A_{0}) and we need to verify F integrability of g on A_{0}. Assume that P = {(x_{1}, A_{1}), ..., (x_{p}, A_{p})} is an ϵ-regular δ_{1}-fine F-partition in A_{0}.

In the next estimations we need the following index sets: I_{1} = {i : x_{i} ∉ cl (A)}, I_{2} = {i : x_{i} ∈ A^{1}∪∂^{∗}A}, I_{3} = {i : x_{i} ∈ cl (A)\(A^{1}∪cl^{∗}A)}, and I_{4} = {i : x_{i} ∈ int^{∗}(A)\A^{1}}. The index sets I_{j} are disjoint and their union is {1, ..., p}. We need to estimate

∑_{i=1}^{p} |g(x_{i})|A_{i}| − F_{0}(A_{i})|.

Split the above sum into four subsums ∑_{i∈I_{j}} for j = 1, ..., 4, (where the empty sum will equal zero by definition).

Step 5. The easy cases: I_{1}, I_{2}.

The sum ∑_{i∈I_{1}} equals 0 since cl (A) ⊃ cl^{∗}(A), and hence for x_{i} ∉ cl (A) we have g(x_{i}) = 0, and by the choice of δ_{1} we also have A_{i} ∩ cl^{∗}(A) = ∅ and hence F_{0}(A_{i}) = 0 as well.

Since δ_{1}(x_{i}) = 0 for x_{i} ∈ A^{1} ∪ ∂^{∗}A the index set I_{2} is empty and hence ∑_{i∈I_{2}} = 0.

Step 6. The case of the index set I_{3}.

Then g(x_{i}) = 0; hence we need to estimate

∑_{i∈I_{3}} |F_{0}(A_{i})| = ∑_{i∈I_{3}} |F (Φ(A_{i} ∩ cl^{∗}A))|.

Here the difficulty is that we cannot find a uniform estimate of the regularity of the sets A_{i} ∩ cl^{∗}A, and of their Φ images. In the next few steps we will use Lemmas 4.1 and 4.2 to solve this problem.

Substep 6.1. The definition of partitions of type P.

Using ϵ from this proof choose η_{0} according to Lemma 4.2. Denote by I_{31} the set of those i ∈ I_{3} for which Φ(A) is d(A_{i})/η_{0}-filling in Q(Φ(x_{i}), η_{0}d(A_{i})) and let I_{32} = I_{3} \ I_{31}. Let P_{11} = {(x_{i}, A_{i})}_{i∈I_{31}}, P_{12} = {(x_{i}, A_{i})}_{i∈I_{32}}. Using Lemma 4.2 choose an F-pre-partition P_{21} = {(Φ(x_{i}), B_{i})}_{i∈I_{31}} which is c_{η_{0}}-equivalent to P_{11} and Φ(A_{j} ∩cl^{∗}(A))∩B_{i} = ∅ for i, j ∈ I_{31}, i ≠ j. Similarly, using Lemma 4.1 with η = η_{0} choose an F-pre-partition P_{22} = {(Φ(x_{i}), B_{i})}_{i∈I_{32}} which is c_{η_{0},ϵ}- equivalent to P_{12} and satisfies B_{i} ∩ Φ(cl^{∗}(A)) = ∅ for i ∈ P_{22}. Observe that this implies Φ(A_{j} ∩ cl^{∗}(A)) ∩ B_{i} = ∅ for i, j ∈ I_{32}, i ≠ j. In the next section we work out in detail the estimation for the partitions P_{11} and P_{21}, the argument for P_{12} and P_{22} is similar (with c_{η_{0}} replaced by c_{η_{0},ϵ}).

Substep 6.2. The estimation for the index set P_{21}. Regularity estimates.

Put B_{i}^{+} = B_{i} ∪ Φ(A_{i} ∩ cl^{∗}(A)), B_{i}^{−} = B_{i} \ Φ(A_{i} ∩ cl^{∗}(A)), P_{21}^{+} = {(Φ(x_{i}), B_{i}^{+})}_{i∈I_{31}}, and P_{21}^{−} = {(Φ(x_{i}), B_{i}^{−})}_{i∈I_{31}}. Next we verify that these partitions are both ϵ_{1}- regular. Since (x_{i}, A_{i}) is ϵ-regular we have |A_{i}| > ϵd(A_{i} ∪ {x_{i}})||A_{i}|| and by (3.1) we also have |A_{i}| ≥ ϵ^{m}d(A_{i} ∪ {x_{i}})^{m}/c_{iso}^{m}. Observe that x_{i} ∈ I_{3} implies that (5.3) is applicable and hence

|Φ(A_{i} ∩ cl^{∗}A)| < c_{Φ}^{m}|A_{i} ∩ cl^{∗}A| < ϵ_{1}c_{Φ}^{m}d(A_{i} ∪ \{x_{i}\})^{m} ≤ ϵ_{1}(\frac{c_{Φ}c_{iso}}{ϵ})^{m}|A_{i}|.

If we choose ϵ_{1} < ϵ^{m}/2c_{η_{0}}c_{Φ}^{m}c_{iso}^{m} then we can continue the above estimation to obtain

|Φ(A_{i} ∩ cl^{∗}A)| < \frac{1}{2c_{η_{0}}} |A_{i}| < \frac{|B_{i}|}{2}

which implies

|B_{i}^{+}| ≥ |B_{i}^{−}| ≥ |B_{i}| − |Φ(A_{i} ∩ cl^{∗}A)| ≥ \frac{|B_{i}|}{2} ≥ \frac{|A_{i}|}{2c_{η_{0}}} .

We also have

d(B_{i}^{−} ∪ {Φ(x_{i})}) ≤ d(B_{i}^{+} ∪ {Φ(x_{i})}) ≤ d(B_{i} ∪ {Φ(x_{i})}) + d(Φ(A_{i} ∩ cl^{∗}A) ∪ {Φ(x_{i})}) ≤

c_{η_{0}}d(A_{i} ∪ {x_{i}}) + c_{Φ}d(A_{i} ∪ {x_{i}}) ≤ (c_{η_{0}} + c_{Φ})d(A_{i} ∪ {x_{i}}).

From (5.2) used with r = d(A_{i} ∪ {x_{i}}) < δ_{1}(x) < δ_{2}(x) it follows that

ϵ^{m−1}d(A_{i} ∪ {x_{i}})^{m−1} > H(∂^{∗}A ∩ B(x_{i}, d(A_{i} ∪ {x_{i}}))) ≥ H(∂^{∗}A ∩ A_{i});

hence

||A_{i} ∩ cl^{∗}A|| ≤ ||A_{i}|| + H(∂^{∗}A ∩ A_{i}) <

||A_{i}|| + ϵ^{m−1}d(A_{i} ∪ \{x_{i}\})^{m−1} < \frac{|A_{i}|}{ϵd(A_{i} ∪ \{x_{i}\})} + \frac{c_{iso}^{m}|A_{i}|}{ϵd(A_{i} ∪ \{x_{i}\})},

where at the last estimate we used the ϵ-regularity of (x_{i}, A_{i}) and (3.1). This implies

||Φ(A_{i} ∩ cl^{∗}A)|| < c_{Φ}^{m−1} (\frac{1 + c_{iso}^{m}}{ϵ} ) \frac{|A_{i}|}{d(A_{i} ∪ \{A_{i}\})}.

Now we have

||B_{i}^{±}|| ≤ ||B_{i}|| + ||Φ(A_{i} ∩ cl^{∗}A)|| ≤ c_{η_{0}}||A_{i}|| + ||Φ(A_{i} ∩ cl^{∗}A)|| ≤

(\frac{c_{η_{0}}}{ϵ} + c_{Φ}^{m−1} \frac{1 + c_{iso}^{m}}{ϵ} ) \frac{|A_{i}|}{d(A_{i} ∪ \{x_{i}\})}.

Therefore combining the above results we have

|B_{i}^{±}| ≥ \frac{|A_{i}|}{2c_{η_{0}}} ≥ \frac{1}{2c_{η_{0}}} (\frac{c_{η_{0}}}{ϵ} + c_{Φ}^{m−1} \frac{1 + c_{iso}^{m}}{ϵ} )^{−1} ||B_{i}^{±}||d(A_{i} ∪ \{x_{i}\}) ≥

\frac{1}{2c_{η_{0}}} (c_{η_{0}} +c_{Φ}^{m−1}(1 + c_{iso})^{m})^{−1} \frac{1}{c_{η_{0}} + c_{Φ}} ϵ||B_{i}^{±}||d(B_{i}^{±}∪\{Φ(x_{i})\}) ≥ c_{2}ϵ||B_{i}^{±}||d(B_{i}^{±}∪\{Φ(x_{i})\})

if c_{2} is a sufficiently small constant depending on ϵ, c_{Φ} and m. This shows that P_{21}^{+} and P_{21}^{−} are both c_{2}ϵ-regular.

Substep 6.3. The estimation for the index set P_{21}. Riemannian sum estimates.

We can assume that 1/c_{2} ≥ c_{η_{0}} + c_{Φ} and hence d(B_{i}^{−} ∪ {Φ(x_{i})}) ≤ d(B_{i}^{+} ∪ {Φ(x_{i})}) ≤ d(A_{i} ∪ {x_{i}})/c_{2}. Recall that A_{i} ⊆ B(x_{i}, δ_{1}(x_{i})), and δ_{1}(x_{i}) ≤ c_{2}δ(Φ(x_{i})); hence d(A_{i} ∪ {x_{i}})/c_{2} < δ(Φ(x_{i})). Therefore B_{i}^{−} ∪ {Φ(x_{i})} ⊆ B_{i}^{+} ∪ {Φ(x_{i})} ⊆ B(Φ(x_{i}), δ(Φ(x_{i}))) ⊆ int (K).

Thus P_{21}^{+} and P_{21}^{−} are both ϵ_{1}-regular δ-fine BV-pre-partitions and hence

∑_{i∈I_{31}} |f(Φ(x_{i}))|B_{i}^{+}| − F (B_{i}^{+})| < ϵ_{1} and ∑_{i∈I_{31}} |f(Φ(x_{i}))|B_{i}^{−}| − F (B_{i}^{−})| < ϵ_{1},

therefore, using B_{i}^{+} \ B_{i}^{−} = Φ(A_{i} ∩ cl^{∗}(A)) we obtain

∑_{i∈I_{31}} |f(Φ(x_{i}))|Φ(A_{i} ∩ cl^{∗}(A))| − F (Φ(A_{i} ∩ cl^{∗}(A)))| < 2ϵ_{1}.

Thus (5.3) used with r = d(A_{i} ∪ {x_{i}}), the Lipschitz properties of Φ, the ϵ-regularity of (x_{i}, A_{i}) and (3.1) imply that there are constants c and c_{3} depending on ϵ, c_{Φ} and m such that

∑_{i∈I_{31}} |F (Φ(A_{i} ∩ cl^{∗}(A)))| < 2ϵ_{1} + ∑_{i∈I_{31}} |f(Φ(x_{i}))| · c · ϵ_{1} \frac{d(A_{i} ∪ \{x_{i}\})^{m}}{|f(Φ(x_{i}))| + 1} ≤

2ϵ_{1} + cϵ_{1} ∑_{i∈I_{31}} d(A_{i} ∪ \{x_{i}\})^{m} ≤ 2ϵ_{1} + cϵ_{1} ∑_{i∈I_{31}} (\frac{c_{iso}}{ϵ} )^{m} |A_{i} ∪ \{x_{i}\}| ≤

2ϵ_{1} + cϵ_{1} (\frac{c_{iso}}{ϵ} )^{m} |A_{0}| < c_{3}ϵ_{1}.

A similar estimation holds for the index set I_{32}. Adding the estimations for I_{31} and I_{32} we can find a constant c_{4} depending on ϵ, c_{Φ} and m such that

∑_{I_{3}} |F_{0}(A_{i})| = ∑_{I_{3}} |F (Φ(A_{i}) ∩ cl^{∗}(A))| < c_{4}ϵ_{1} < ϵ,

where for the last estimate we can assume that ϵ_{1}, depending on ϵ, c_{Φ} and m was chosen sufficiently small.

Step 7. The case of the index set I_{4}.

Put B_{i} = Φ(A_{i} ∩ cl^{∗}A). By the choice of δ_{1} we have B_{i} ⊆ B(Φ(x_{i}), c_{Φ}δ_{1}(x_{i})) ⊆ B(Φ(x_{i}), δ(Φ(x_{i}))) ⊆ int (K).

Inequalities (5.4) and δ_{3} > δ_{1} > r = d(A_{i} ∪{x_{i}}) imply |B(x_{i}, d(A_{i} ∪{x_{i}}))\cl^{∗}(A)| < ϵ_{1}d(A_{i} ∪ {x_{i}})^{m} < ϵd(A_{i} ∪ {x_{i}})^{m}, (5.2) and δ_{2} > δ_{1} imply H(∂^{∗}A ∩ B(x_{i}, d(A_{i}))) < ϵ^{m−1}d(A_{i} ∪ {x_{i}})^{m−1}. Recall that (x_{i}, A_{i}) is ϵ-regular. Clearly, there exists a constant c̃ > 0 depending only on m such that if ϵ < c̃ then the previous two estimations imply that |A_{i} ∩ cl^{∗}A| > |A_{i}|/2 and ||A_{i} ∩ cl^{∗}A|| < 2||A_{i}||. Without limiting generality we can assume that ϵ < c̃. Since {(x_{i}, A_{i})}_{i∈I_{4}} is ϵ-regular and Φ is a lipeomorphism there is a constant c_{5} > 0 depending on c_{Φ} and m such that {(Φ(x_{i}), B_{i})}_{i∈I_{4}} is a c_{5}ϵ-regular partition. We can also assume that ϵ_{1} is so small that c_{5}ϵ > ϵ_{1}.

For x_{i} ∈ int^{∗}(A) \ A^{2} by the choice of δ_{4}(x_{i}) > δ_{1}(x_{i}) we can use (5.5) with B_{0} = A_{i}. Since A^{2} is of measure zero and one cannot change integrability of a function by changing it on a set of measure zero we can assume that g(x) = f(Φ(x)) = 0 for x ∈ A^{2}.

Denote by I_{41} the set of those i’s in I_{4} for which x_{i} ∉ A^{2}. We also put I_{42} = I_{4} \ I_{41}. Then

∑_{I_{4}} |g(x_{i})|A_{i}| − F_{0}(A_{i})| = ∑_{I_{41}} |g(x_{i})|A_{i}| − F_{0}(A_{i})| + ∑_{I_{42}} |g(x_{i})|A_{i}| − F_{0}(A_{i})|.

Using (5.4) with r = d(A_{i} ∪ {x_{i}}), x = x_{i}, (5.5) with B_{0,i} = A_{i} ∩ cl^{∗}A, Φ(B_{0,i}) = Φ(A_{i} ∩ cl^{∗}A) = B_{i}, x = x_{i} for i ∈ I_{41} and (5.1) we obtain that there exists a constant c_{6} depending on c_{Φ} and m such that

∑_{I_{41}} |g(x_{i})|A_{i}| − F_{0}(A_{i})| = ∑_{I_{41}} |f(Φ(x_{i}))J_{Φ}(x_{i})|A_{i}| − F (Φ(A_{i} ∩ cl^{∗}A))| ≤

∑_{I_{41}} |f(Φ(x_{i}))| · (|J_{Φ}(x_{i})|A_{i}| − J_{Φ}(x_{i})|A_{i} ∩ cl^{∗}A|| + |J_{Φ}(x_{i})|A_{i} ∩ cl^{∗}A| − |B_{i}||) +

∑_{I_{41}} |f(Φ(x_{i}))|B_{i}| − F (B_{i})| ≤ (∑_{I_{41}} ϵ_{1}d(A_{i} ∪ {x_{i}})^{m} + c_{5}ϵ|A_{i} ∩ cl^{∗}A|) + ϵ_{1} ≤

(∑_{I_{41}} ϵ_{1} (\frac{c_{iso}}{ϵ} )^{m} |A_{i} ∪ \{x_{i}\}| + c_{5}ϵ|A_{i}|) + ϵ_{1} ≤ c_{6}ϵ max\{1, |A_{0}|\},

where at the last estimate we also assumed that ϵ_{1} was chosen so small that ϵ_{1}(c_{iso}/ϵ)^{m} < ϵ. On the other hand,

∑_{I_{42}} |g(x_{i})|A_{i}| − F_{0}(A_{i})| = ∑_{I_{42}} |F (Φ(A_{i} ∩ cl^{∗}A))| =

∑_{I_{42}} |F (B_{i})| = ∑_{I_{42}} |f(Φ(x_{i}))|B_{i}| − F (B_{i})| < ϵ_{1} < ϵ.

Therefore there exists a constant c_{7} depending on c_{Φ} and m such that

∑_{j=1}^{4} ∑_{i∈I_{j}} |g(x_{i})|A_{i}| − F_{0}(A_{i})| = ∑_{i=1}^{p} |g(x_{i})|A_{i}| − F_{0}(A_{i})| < c_{7}ϵ.

This completes the proof of the F integrability of (f ∘ Φ)J_{Φ} on A_{0}. From what we verified it follows that BVF ∫_{A}(f ∘ Φ)J_{Φ} = F_{0}(A_{0}) = F (Φ(A_{0} ∩ cl^{∗}A)) = F (Φ(A)) = BVF ∫_{Φ(A)} f, this is the change of variables formula for the BVF integral.

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Eötvös Loránd University, Department of Analysis, Budapest, Múzeum krt 6-8, H-1088, Hungary.

email: buczo@cs.elte.hu