absolutely continuous variational measures of mawhin s type

5, 747–768 ABSOLUTELY CONTINUOUS VARIATIONAL MEASURES OF MAWHIN’S TYPE Francesco Tulone* — Yurij Zherebyov** Communicated by L’ubica Hol´ a ABSTRACT.. We obtain characterization of thes

DOI: 10.2478/s12175-011-0043-0

Math Slovaca 61 (2011), No 5, 747–768

ABSOLUTELY CONTINUOUS VARIATIONAL MEASURES OF MAWHIN’S TYPE

Francesco Tulone* — Yurij Zherebyov**

(Communicated by L’ubica Hol´ a )

ABSTRACT In this paper we study absolutely continuous and σ-finite

vari-ational measures corresponding to Mawhin, F - and BV -integrals We obtain

characterization of theseσ-finite variational measures similar to those obtained

in the case of standard variational measures We also give a new proof of the Radon-Nikod´ ym theorem for these measures.

c

2011

Mathematical Institute Slovak Academy of Sciences

1 Introduction

Various variational measures have been actively studied during recent years

in connection with the problem of descriptive characterization of conditionally

convergent integrals It turned out that the absolute continuity and σ-finiteness

play the central role in the theory of these measures Absolutely continuousvariational measures characterize primitives of conditionally convergent integrals(see, for example, [3, 4, 6, 7, 8, 9, 14, 15, 17, 18, 27, 28, 31, 33, 36, 37, 38, 39, 40,

45, 48, 49]) At the same time, σ-finiteness of a variational measure gives some

information about differentiability properties of the set function that determinesthis measure (see [2, 4, 6, 7, 10, 13, 21, 24, 32, 35, 36, 40, 41, 42, 44, 49]) It

is also well-known that, unlike general situation, the absolute continuity of a

variational measure implies its σ-finiteness (see [3, 4, 6, 7, 9, 12, 14, 15, 17, 19, 27,

31, 33, 36, 39, 40, 44, 46, 47, 49]) This relation between the absolute continuity

and σ-finiteness motivated several authors to find characterizations of σ-finite

variational measures In general, these characterizations can be expressed in the

following form: variational measure is σ-finite on a set E iff it is σ-finite on

2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 28A12, 26A39.

K e y w o r d s: derivation basis, variational measure, Mawhin integral,F -integral, BV -integral,

Radon-Nikod´ ym derivative.

Research of the second author was supported by RFBR (grant no 08-01-00669).

each set B ⊂ E which is “small” in some sense (see [6, 7, 9, 39, 44, 47, 49]).

“Small” can mean negligible, Borel negligible, compact negligible, etc But thecase of Mawhin’s type variational measures still has not been considered In thispaper we give similar characterizations for several variational measures of thistype (see Theorem 2 and Corollary 1)

Unlike general situation, the Radon-Nikod´ym derivative of several variationalmeasures can be found in the explicit form (see [2, 5, 10, 13, 15, 17, 27, 32, 33,

42, 43, 45, 49]) For Mawhin’s type variational measures generated by charges

it was done by B Bongiorno, Z Buczolich, W F Pfeffer and B S Thomson

in [10, Theorem 3.3, Proposition 4.7], [13, Proposition 4.2, Corollary 4.8] and[32, Proposition 3.2, Theorem 3.6] Their proofs use the concept of essentialvariation introduced by B Bongiorno and P Vetro ([2, 11]) and the result of

B Bongiorno [2, Theorem 1] on the Radon-Nikod´ym derivative of essential ation (see also [5, Theorem 1]) In this paper we also give a new proof of thelatter result (see Theorems 3, 4, 5) This proof is based on Fatou’s lemma

called P-adic intervals A set E ⊂ R m is called a figure if it is a finite union

of intervals With E, int E and ∂E we shall denote the closure, the interior and the boundary of a set E, respectively By d(x, y) and xmaxwe denote the

Euclidean metric and the maximum norm in Rm; i.e.,

d(x, y) =

m i=1 (x i − y i)2

is called the diameter of a set E in the maximum

norm By B(x, R) and U (x, R) we denote open balls of the radius R centered

at x in the Euclidean metric and in the maximum norm, respectively H , µ

and µ ∗ denote the (m − 1)-Hausdorff measure, the Lebesgue measure and the

outer Lebesgue measure in Rm, respectively Terms “measurable” and “almosteverywhere” will always be used in the sense of the Lebesgue measure A set

E ⊂ R m is negligible if µ(E) = 0 Unless specified otherwise, the absolute

conti-nuity of a measure we understand in with respect to the Lebesgue measure For

a measurable set E define the essential interior int ∗ E as the set of all density

points of E and the essential closure cl ∗ E as the set of all nondispersion points

of E (for definitions of density and dispersion points see [1, 29, 34]) Then the set ∂ ∗ E = (cl ∗ E) \ (int ∗ E) is called the essential boundary of E It is easy to see

that the following inclusions hold: int E ⊂ int ∗ E ⊂ cl ∗ E ⊂ E and ∂ ∗ E ⊂ ∂E.

The value E = H (∂ ∗ E) is called the perimeter of a measurable set E A

bounded set E is said to be a BV -set if its perimeter E is finite A set E of

σ-finite Hausdorff measure H is called thin It is easy to check that each thin

set is negligible, but not vice versa Measurable sets A and B are nonoverlapping

if µ(A ∩ B) = 0.

Two standard indicators are usually used to get information how much a

bounded measurable set differs from the m-dimensional cube For a nonempty bounded measurable set E the numbers

reg(E) =

 µ(E)

(diam E)·E , if (diam E) · E > 0;

are called the shape and the regularity of E (see [10, 12, 13, 16, 29, 30, 31, 32]).

In view of [29, Proposition 12.1.6], shape and regularity are related with theinequality

reg(E)
1

2m [ρ(E)]

1

for any figure E Thus, if E is a figure, it is easy to check that 0 < ρ(E)
1,

0 < reg(E)
2m1 and m-dimensional cube is a figure of maximum shape 1 and

regularity 2m1 If E is an interval then the opposite inequality holds; i.e.,

1

(see [29, Remark 12.1.7])

Let Ψ be a class of bounded measurable sets Fix a positive function δ on R m

and consider the set

• be a Perron basis if (x, M) ∈ B δ implies x ∈ M for each δ;

• be a Vitali basis if B δ[{x}] = ∅ holds for each x and for each δ;

• be a BF -basis if for any B-set M and for any x ∈ M the pair (x, M) ∈ B δ for some positive function δ;

• have the Vitali property if for any set E ⊂ R m and for any collection of

B-sets C , which forms a Vitali cover of E, there exists a subsystem {M i } ∞

i=1 ⊂C of pairwise disjoint B-sets M i such that µ ∗

E \

∞ i=1

If all the setsB ρδ andB rδ are nonempty for each positive function δ, then they

clearly form derivation bases B ρ ={B ρδ } δ and B r = {B rδ } δ In this case wesay that the basisB generates regular bases B ρandB r Throughout this paper

we will use for regular derivation bases the lower index ρ, if we mean shape, and

r if we mean regularity.

LetB be a Vitali derivation basis with all B-sets of positive Lebesgue

mea-sure The upper and the lower B-derivates at a point x are defined as

D B F (x) = inf

δ sup



F (M ) µ(M ) : (x, M ) ∈ B δ[{x}]

AB-set function F is said to be B-differentiable at a point x if both extreme B-derivates at x are finite and coincide Their common value is called the B-derivative of F at x and is denoted D B F (x).

Let, in addition, all regular derivation bases B r , r ∈

0, 2m1 , generated bythe basisB, be Vitali bases Then the upper and the lower ordinary B-derivates

of aB-set function F at a point x are defined in the following way:

F  B (x) = sup

r D B r F (x) and F  B (x) = inf

r D B r F (x).

If both extreme ordinary B-derivates at a point x are finite and coincide, then

the function F is B-differentiable in the ordinary sense at x Their common value is called the ordinary B-derivative of F at x and is denoted F 

A B-set function is additive if F (L ∪ M) = F (L) + F (M) for each pair of

nonoverlappingB-sets L and M A derivation basis B has the Ward property,

whenever each additiveB-set function F is B-differentiable at almost all points

at which both extremeB-derivates are finite.

If in the definition of derivation basis, which is supposed to be Perron, we put

Ψ to be the family of all intervals,P-adic intervals, figures or BV -sets, we will

get the full basis B K H , the P-adic basis B P , the basis of figures F or the basis of BV -sets BV , respectively Regular bases B ρ K H, B K H

ρ , B P

r ,

F ρ, F r, BV ρ and BV r are defined in a natural way It is clear that all these

bases are Vitali and BF -bases By the Vitali covering theorem ([34, Ch 4,

§3, Theorem 3.1]) and by (1), the bases B K H

ρ , B P

r , F ρ and

F r have the Vitali property Moreover, by Ward’s theorems ([34, Ch 4,§11])

and by [8, Theorem 4.1], the bases B K H, B K H

r withbounded sequence P, ρ ∈ (0, 1) and r ∈ (0, 1

2m), have the Ward property Aweakened analogue of Ward’s theorem holds for the basesF and BV , too (see

[13, Lemma 3.1, Theorem 3.3])

The use of gages is the central idea of the whole theory of nonabsolutelyconvergent integrals and the theory of variational measures The term ‘gage’, asmet in the literature, has several meanings The original and standard definition

of gage is due to J Kurzweil and R Henstock (see [22, 23, 25]) However, somemodifications were recently suggested by a number of authors Here we considerthe gage in a general setting which covers all the most important cases Let

an arbitrary family K of negligible sets be fixed A nonnegative real-valued

function δ(·) is called a gage if its null set 

x : δ(x) = 0

∈ K If we put

K = {∅}, then we obtain the classical gage which is used in the theory of

Henstock-type integrals (see, for example, [6, 8, 17, 21, 26, 27, 28, 33, 39, 43, 44])

If we putK to be the family of all thin sets, then we get the gage which is used

in the study ofF - and BV -integrals (see, for example, [3, 10, 12, 13, 14, 15, 16,

29, 30, 31, 32]) At last, ifK is the family of all negligible sets, we obtain the

notion of essential gage (see, for example, [3, 5, 10, 13, 32]).

Let B be a Vitali derivation basis, W ⊂ R m , F be a B-set function and

δ be a gage defined on W A finite collection π ⊂ B δ [E] is a δ-fine partition

anchored in E ⊂ W , if for any pairs (x, M ), (y, L) ∈ π the sets M and L are

nonoverlapping The set function

where π is δ-fine partition anchored in E, is called the δ-variation of F on E.

Then, the set functions

V (K , B, F, E) = inf

δ Var(K , B δ , F, E) (4)and

generated by F with respect to the basis B (or simply the variational measure

of Mawhin’s type) Inequalities (1) and (2) show that in the case of derivation

bases of intervals the variational measure of Mawhin’s type allows an equivalentdefinition in terms of shape; i.e.,

B, F, ·, respectively It is easy to check that V (B, F, ·) and V M(B, F, ·)

are metric outer measures, so by Caratheodory’s theorem (see [1, V 1, rem 1.11.9], [20, Ch 1,§ 1.1, Theorem 5]) they are Borel measures in the metric

Theo-space (W, d) Unless specified otherwise, by σ-finiteness of variational measures

V (B, F, ·) and V M(B, F, ·) we understand their σ-finiteness as Borel measures.

It follows directly from the definitions that

A We say that a set A ∈ A is regular with respect to the class S and outer measure m if there exists a set C ∈ S which contains A and m(A) = m(C) The

notion of regularity of outer measure is often used in the case when A is the σ-algebra of all subsets of a fixed set and S is the σ-algebra of all measurable

sets in the sense of Caratheodory (see, for example, [1, 20, 42])

A real-valued BV -set function F is called a charge if it satisfies the following

It is known that the family of all BV -sets can be topologized so that each

charge becomes a continuous function with respect to this topology (see [10,12], [13, Remark 1.2], [15, 16], [29, Remark 13.2.20], [30, Remark 2.3] and [32,Remark 1.2])

3 σ-finite variational measures

We need the following theorem which extends [9, Lemma 3.1], [27, rem 3.7], [45, Lemma 4] for the case of an abstract measure space

Theo-
 1 Let G, A be σ-algebras of subsets of a set E and G ⊂ A Let λ

be a σ-finite and σ-additive measure defined on A Let ν be an outer measure defined on A , with the restriction on σ-algebra G being σ-additive Assume also that each set from A is regular with respect to σ-algebra G and measure λ If the outer measure ν is absolutely continuous with respect to the measure λ, then

ν is a σ-additive measure on A

P r o o f First we prove that each set Y ∈ A contains a set X ∈ G such that

λ(Y \ X) = 0 By σ-finiteness of λ the following representation holds: E =

+∞

k=1

E k , where E k ∈ A and λ(E k ) < +∞ (k = 1, 2, 3, ) Then the set C k =

(E \ Y ) ∩ E k ∈ A By regularity there exists a set V k ∈ G containing C k, such

that λ(V k ) = λ(C k)
λ(E k ) < +∞ Then it is easy to check that a required set is X = E \

+∞

k=1

V k

Now we prove the theorem Let A1, A2, A3, be pairwise disjoint sets from

A As it was proved above, there are sets X n ∈ G such that X n ⊂ A n and

λ(A n \ X n ) = 0 (n = 1, 2, 3, ) Then by the absolute continuity we have

ν(A n \ X n ) = 0 (n = 1, 2, 3, ) Using σ-additivity of ν on σ-algebra G we get

 2 Let B be a Perron and Vitali derivation BF -basis defined on a measurable set E ⊂ R m , satisfying the following conditions:

a) each B-set M has nonempty interior int M;

b) each point x of a B-set M , belonging to the boundary ∂M , is the point of

positive lower density for int M

Let F be a B-set function Then the variational measure V

{∅}, B, F, · is σ-finite on E iff it is σ-finite on each negligible set B ⊂ E which is a G δ -set

in E.

In particular, absolutely continuous variational measure V

{∅}, B, F, · is σ-finite σ-additive measure defined on σ-algebra of all measurable subsets of E.

P r o o f The necessity is clear We prove the sufficiency By measurability of

E there exists an F σ -set P ⊂ E with µ(E \ P ) = 0 Then by [47, Theorem 1] the variational measure V

{∅}, B, F, · is σ-finite on P There also exists a negligible G δ -set G ⊃ (E \ P ) Then the set E ∩ G is a G δ -set in E and

µ(E ∩ G) = 0 Hence, by the assumption of lemma the variational measure

ν is the variational measure V

{∅}, B, F, · and λ is the Lebesgue measure µ.

3 Let K be an arbitrary class of negligible subsets of a measurable set

E ⊂ R m and F be a charge If the variational measure V M

P r o o f For classesK having the property:

if B ∈ K and C is at most countable subset of E, then B ∪ C ∈ K , (7) Lemma 3 is a special case of [30, Theorem 3.3] In fact, (7) means: if δ is a gage

on A ⊂ R m, then the function

δ(x) =



δ(x), if x ∈ A \ C;

0, if x ∈ C

is a gage on A, too In the general case, the proof given in [30, Theorem 3.3]

requires a slight modification

Since each figure is a BV -set, by definition

V M

K , F , F, A
V M

K , BV , F, A for each set A ⊂ R m

Let κ be the positive constant used in [30, Proposition 2.4] Proceeding towards

a contradiction, suppose that there exists a set B ⊂ R m such that

V M

K , F , F, B< V M

K , BV , F, B.

It means that there is a regularity r ∈

0, 2m1 and a gage δ : B → [0, +∞) such

Since the variational measure V

Applying the construction given in the proof of [30, Theorem 3.3], we get from

the partition π a partition Π ⊂ F θδ [B \ C] such that

2 Let F be a charge Then the variational measure V M

P r o o f The necessity is clear Prove the sufficiency Consider an arbitrarysequence

r n

+∞

n=1 monotone convergent to zero with r1∈0, 2m1 

It is easy tosee that bases F r n generated by F satisfy conditions of Lemma 2 Moreover,

by the Vitali covering theorem [34, Ch 4,§ 3, Theorem 3.1] and by (1) all these

bases have the Vitali property It follows from (6) that variational measures

V

{∅}, F r n , F, ·

are σ-finite on each negligible set B ⊂ E which is a G δ-set in

E (n = 1, 2, 3, ) Then by Lemma 2 variational measures V

{∅}, F r n , F, ·

are σ-finite on E Let C n = 

x ∈ E : D F rn |F |(x) < +∞ Since r n-regularderivatesD F rn |F | are measurable (see [34, Ch 4, § 4, Theorem 4.2]), sets C narealso measurable and by Lemma 1

µ(E \ C n) = 0 (n = 1, 2, 3, ).

Let C =

+∞

n=1

C n All r n -regular derivates D F rn |F | are finite at each point of C.

Hence, in view of r n → 0 and by (3), all upper derivates D F r |F |r ∈ (0, r1]

Fix an arbitrary r ∈

0, 2m1 

By (3) the r-regular derivative D F r F exists at

each point x ∈ P k and D F r F (x)=F 

F (x)< k Then for each x ∈ P k thereexists δ(x) > 0 such that for each pair (x, M ) ∈ F r
δ[{x}],

i=1 ⊂ F rδ [P k ] Since the figures M i are

nonoverlap-ping, by the definition of the gage δ

There exists a

negligible G δ -set H ⊃ Z The set B = E ∩ H is a negligible G δ -set in E Then, by assumption of the theorem, the variational measure V M

{∅}, F , F, ·

is σ-finite on B Moreover, the following representation holds E =

∞ k=1

 1 Let B be a Perron and Vitali derivation BF -basis defined on

a measurable set E ⊂ R m with B-sets being unions of intervals Moreover, let regular bases B1

n , generated by B, have Vitali and Ward properties for each

n ( n1 denotes here either the shape or regularity) Let F be an additive B-set function Then the variational measure V M

{∅}, B, F, ·is σ-finite on E iff it

is σ-finite on each negligible set B ⊂ E which is a G δ -set in E.

In particular, absolutely continuous variational measure V M

{∅}, B, F, · is σ-finite σ-additive measure defined on the σ-algebra of all measurable subsets

of E.

In particular, in view of [8, Theorem 4.1] and [34, Ch 4, § 11], the bases

B K H andB Pwith bounded sequenceP satisfy conditions of the Corollary 1.

Remark The question, whether Theorem 2 holds for an arbitrary class K , is

open The basisBV does not satisfy conditions of Corollary 1 Therefore, the

question, whether Theorem 2 holds for the variational measure V M

K , BV, F, ·,also remains open

The classical Radon-Nikod´ym theorem applied to variational measures can

be refined It is possible to find the explicit formula for the Radon-Nikod´ymderivative of some variational measures (see [2, 5, 10, 13, 15, 17, 32, 33, 42, 43,49]) In this section we will give a new proof of these results Theorem 3 playsthe central role (see [2, Theorem 1] for original proof) Here we give a new proof

of this theorem based on Fatou’s lemma For similar results in dimension onesee also [5, Theorem 1], [33, Proposition 10], [42, Corollary 7.11] and [43]

 3 Let B be a Perron and Vitali derivation basis defined on a surable set E ⊂ R m with the Vitali property, with all B-sets being unions of in- tervals, and let F be a B-set function Then the variational measure V

mea-B, F, ·

get the full basis B K H , the P-adic basis B P , the basis of. .. of figures F or the basis of BV -sets BV , respectively Regular bases B ρ K H, B K H

ρ , B P...

Theo-space (W, d) Unless specified otherwise, by σ-finiteness of variational measures< /i>

V (B, F, ·) and V M(B, F, ·) we understand their σ-finiteness as