absolutely continuous variational measures of mawhin s type

Ó 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 variational measures corresponding to Mawhin, F - and BV -integrals We obtain characterization of these σ-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 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 continuous variational 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 determines this 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) FRANCESCO TULONE — YURIJ ZHEREBYOV 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 the case of Mawhin’s type variational measures still has not been considered In this paper we give similar characterizations for several variational measures of this type (see Theorem and Corollary 1) Unlike general situation, the Radon-Nikod´ ym derivative of several variational measures 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 essential variation 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 variation (see also [5, Theorem 1]) In this paper we also give a new proof of the latter result (see Theorems 3, 4, 5) This proof is based on Fatou’s lemma Preliminaries An interval will always be m-dimensional compact interval in Rm Let P = {pi }∞ i=1 be a sequence of natural numbers Put l0 = 1, ln = ln−1 · pn (n = +1 1, 2, 3, ) Intervals ldi1 , d1li+1 × · · · × ldim , dm (dk lim m 1 m called P-adic intervals A set E ⊂ R is called a figure of intervals With E, int E and ∂E we shall denote the and the boundary of a set E, respectively By d(x, y) and Euclidean metric and the maximum norm in Rm ; i.e., m (xi − yi ) d(x, y) = i=1 2 and x max = 0, , lik − 1) are if it is a finite union closure, the interior x max we denote the = max |xi |, i=1, ,m where x = (x1 , , xm ), y = (y1 , , ym ) ∈ Rm The value diam E = sup x − y max : x, y ∈ E 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 “almost everywhere” will always be used in the sense of the Lebesgue measure A set E ⊂ Rm is negligible if µ(E) = Unless specified otherwise, the absolute continuity 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 748 ABSOLUTELY CONTINUOUS VARIATIONAL MEASURES OF MAWHIN’S TYPE 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) = 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 ρ(E) = 0, if E is a point; otherwise µ(E) (diam E)m , and reg(E) = µ(E) (diam E)· E , 0, if (diam E) · E > 0; otherwise 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 the inequality 1 reg(E) (1) [ρ(E)] m 2m for any figure E Thus, if E is a figure, it is easy to check that < ρ(E) 1, < reg(E) 2m and m-dimensional cube is a figure of maximum shape and regularity 2m If E is an interval then the opposite inequality holds; i.e., [ρ(E)] 2m reg(E) (2) (see [29, Remark 12.1.7]) Let Ψ be a class of bounded measurable sets Fix a positive function δ on Rm and consider the set Bδ = (x, M ) : M ⊂ B(x, δ(x)), M ∈ Ψ The family B = {Bδ }δ , where each Bδ is nonempty and δ runs over the set of all positive functions, is called a derivation basis Elements of the class Ψ are called B-sets We will also use the following notation Bδ [E] = (x, M ) ∈ Bδ : x ∈ E for a set E ⊂ R The following definitions are taken from [6, 17, 39, 47, 49] A derivation basis B is said to m 749 FRANCESCO TULONE — YURIJ ZHEREBYOV • be a Perron basis if (x, M ) ∈ Bδ implies x ∈ M for each δ; • be a Vitali basis if Bδ [{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 ) ∈ Bδ for some positive function δ; • have the Vitali property if for any set E ⊂ Rm and for any collection of B-sets C , which forms a Vitali cover of E, there exists a subsystem ∗ {Mi }∞ i=1 ⊂ C of pairwise disjoint B-sets Mi such that µ E \ ∞ Mi = i=1 and for each positive Let B be a derivation basis Fix ρ ∈ (0, 1], r ∈ 0, 2m m function δ on R set Bρδ = (x, M ) ∈ Bδ : ρ(M ) ρ and Brδ = (x, M ) ∈ Bδ : reg(M ) r If all the sets Bρδ and Brδ are nonempty for each positive function δ, then they clearly form derivation bases Bρ = {Bρδ }δ and Br = {Brδ }δ In this case we say that the basis B generates regular bases Bρ and Br Throughout this paper we will use for regular derivation bases the lower index ρ, if we mean shape, and r if we mean regularity Let B be a Vitali derivation basis with all B-sets of positive Lebesgue measure The upper and the lower B-derivates at a point x are defined as D B F (x) = inf sup F (M ) µ(M ) : (x, M ) ∈ Bδ [{x}] D B F (x) = sup inf F (M ) µ(M ) : (x, M ) ∈ Bδ [{x}] δ and δ A B-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 DB F (x) , generated by Let, in addition, all regular derivation bases Br , r ∈ 0, 2m the basis B, be Vitali bases Then the upper and the lower ordinary B-derivates of a B-set function F at a point x are defined in the following way: F B (x) = sup D Br F (x) r and F B (x) = inf D Br F (x) r 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 FB (x) It follows from inequalities (1) and (2) that, for derivation bases of intervals this definition of the ordinary derivative is equivalent to the classical one given in [34] Provided < r1 r2 2m , it is trivial that F B (x) 750 DBr F (x) D Br F (x) D Br2 F (x) DBr1 F (x) F B (x) (3) ABSOLUTELY CONTINUOUS VARIATIONAL MEASURES OF MAWHIN’S TYPE A B-set function is additive if F (L ∪ M ) = F (L) + F (M ) for each pair of nonoverlapping B-sets L and M A derivation basis B has the Ward property, whenever each additive B-set function F is B-differentiable at almost all points at which both extreme B-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 BρK H , BrK H , BρP , BrP , Fρ , Fr , 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 BρK H , BrK H , BρP , BrP , Fρ and Fr have the Vitali property Moreover, by Ward’s theorems ([34, Ch 4, §11]) and by [8, Theorem 4.1], the bases B K H , BρK H , BrK H , BρP and BrP with ), have the Ward property A bounded sequence P, ρ ∈ (0, 1) and r ∈ (0, 2m weakened analogue of Ward’s theorem holds for the bases F and BV , too (see [13, Lemma 3.1, Theorem 3.3]) The use of gages is the central idea of the whole theory of nonabsolutely convergent integrals and the theory of variational measures The term ‘gage’, as met 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, some modifications were recently suggested by a number of authors Here we consider the 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) = ∈ 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 put K to be the family of all thin sets, then we get the gage which is used in the study of F - and BV -integrals (see, for example, [3, 10, 12, 13, 14, 15, 16, 29, 30, 31, 32]) At last, if K 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 ⊂ Rm , F be a B-set function and δ be a gage defined on W A finite collection π ⊂ Bδ [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 Var(K , Bδ , F, E) = sup π |F (M )|, (x,M )∈π 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 Var(K , Bδ , F, E) δ (4) and 751 FRANCESCO TULONE — YURIJ ZHEREBYOV VM (K , B, F, E) = sup r∈(0, 2m ] V (K , Br , F, E) (5) are called the variational measure generated by F with respect to the basis B (or simply the variational measure) and the variational measure of Mawhin’s type 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 equivalent definition in terms of shape; i.e., VM (K , B, F, E) = sup V (K , Bρ , F, E) ρ∈(0,1] Unless specified otherwise, all conclusions about variational measures will be considered under the assumption that all members of K are negligible Variational measures (4) and (5) in this case will be denoted simply as V B, F, · and VM B, F, · , respectively It is easy to check that V (B, F, ·) and VM (B, F, ·) are metric outer measures, so by Caratheodory’s theorem (see [1, V 1, Theorem 1.11.9], [20, Ch 1, § 1.1, Theorem 5]) they are Borel measures in the metric space (W, d) Unless specified otherwise, by σ-finiteness of variational measures V (B, F, ·) and VM (B, F, ·) we understand their σ-finiteness as Borel measures It follows directly from the definitions that V Br2 , F, E V Br1 , F, E VM B, F, E , (6) provided E ⊂ Rm and < r1 r2 2m Let m be an outer measure defined on a σ-algebra A and S be a subclass of 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 conditions: 1) F is additive; 2) for each ε > there exists η > such that |F (B)| < ε for any BV -set B ⊂ U 0, 1ε with µ(B) < η and B < 1ε 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]) 752 ABSOLUTELY CONTINUOUS VARIATIONAL MEASURES OF MAWHIN’S TYPE σ-finite variational measures We need the following theorem which extends [9, Lemma 3.1], [27, Theorem 3.7], [45, Lemma 4] for the case of an abstract measure space Ì ĨƯ Đ 1º 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) = By σ-finiteness of λ the following representation holds: E = +∞ Ek , where Ek ∈ A and λ(Ek ) < +∞ (k = 1, 2, 3, ) Then the set Ck = k=1 (E \ Y ) ∩ Ek ∈ A By regularity there exists a set Vk ∈ G containing Ck , such that λ(Vk ) = λ(Ck ) λ(Ek ) < +∞ Then it is easy to check that a required set is X = E \ +∞ Vk k=1 Now we prove the theorem Let A1 , A2 , A3 , be pairwise disjoint sets from A As it was proved above, there are sets Xn ∈ G such that Xn ⊂ An and λ(An \ Xn ) = (n = 1, 2, 3, ) Then by the absolute continuity we have ν(An \ Xn ) = (n = 1, 2, 3, ) Using σ-additivity of ν on σ-algebra G we get +∞ ν +∞ An +∞ ν(Xn ) + ν(An \ Xn ) ν An n=1 n=1 +∞ = n=1 +∞ ν Xn = ν n=1 +∞ Xn n=1 ν An n=1 This proves σ-additivity of ν on σ-algebra A Analyzing proofs of special results [21, Theorem 3.10] and [44, Theorem 2] it is easy to deduce the same statement for the case of an abstract derivation basis in Rm Ä ÑÑ 1º Let a derivation basis B be defined on a measurable set E ⊂ Rm Assume that B is a Perron and Vitali basis with the Vitali property Let F be a B-set function If the variational measure V B, F, · is σ-finite on E, then µ x ∈ E : D B |F |(x) = +∞ = For a complete proof of Lemma see [49, Ch 2, § 2.2, Lemma 2.2] 753 FRANCESCO TULONE — YURIJ ZHEREBYOV Ä ÑÑ 2º Let B be a Perron and Vitali derivation BF -basis defined on a measurable set E ⊂ Rm , 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 Gδ -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 Fσ -set P ⊂ E with µ(E \ P ) = Then by [47, Theorem 1] the variational measure V {∅}, B, F, · is σ-finite on P There also exists a negligible Gδ -set G ⊃ (E \ P ) Then the set E ∩ G is a Gδ -set in E and µ(E ∩ G) = Hence, by the assumption of lemma the variational measure V {∅}, B, F, · is σ-finite on E∩G Since E = P ∪(E∩G) and P, E∩G are Borel sets in E, the variational measure V {∅}, B, F, · is σ-finite on E σ-additivity of absolutely continuous variational measure V {∅}, B, F, · on σ-algebra of all measurable subsets of E follows from Theorem 1, provided A is the σ-algebra of all measurable subsets of E, G is the Borel σ-algebra of the metric space (E, d), ν is the variational measure V {∅}, B, F, · and λ is the Lebesgue measure µ The proof is complete Ä ÑÑ 3º Let K be an arbitrary class of negligible subsets of a measurable set E ⊂ Rm and F be a charge If the variational measure VM K , BV , F, {x} = at each point x ∈ E then VM K , BV , F, A = VM K , F , F, A for each set A ⊂ Rm P r o o f For classes K having the property: if B ∈ K and C is at most countable subset of E, then B ∪ C ∈ K , (7) Lemma is a special case of [30, Theorem 3.3] In fact, (7) means: if δ is a gage on A ⊂ Rm , then the function δ(x) = δ(x), 0, if x ∈ A \ C; 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 VM K , F , F, A 754 VM K , BV , F, A for each set A ⊂ Rm ABSOLUTELY CONTINUOUS VARIATIONAL MEASURES OF MAWHIN’S TYPE Let κ be the positive constant used in [30, Proposition 2.4] Proceeding towards a contradiction, suppose that there exists a set B ⊂ Rm such that VM K , F , F, B < VM K , BV , F, B and a gage δ : B → [0, +∞) such It means that there is a regularity r ∈ 0, 2m that r Var K , Fθδ , F, B < V K , BV r , F, B , where θ = (8) 2κ By [30, Lemma 3.2] there exists at most countable set C ⊂ B such that the function δ(x), if x ∈ B \ C; δ(x) = 0, if x ∈ C attains strict local maximum at no point x ∈ B Since the variational measure V K , BV r , F, · is an outer measure, by the assumption of the lemma (8) Var K , Fθδ , F, B < V K , BV r , F, B V K , BV r , F, B \ C + V K , BV r , F, C V K , BV r , F, B \ C + VM K , BV , F, C = V K , BV r , F, B \ C Now one can find a partition π ⊂ BV rδ/2 [B \ C] such that Var K , Fθδ , F, B < |F (M )| (x,M )∈π Applying the construction given in the proof of [30, Theorem 3.3], we get from the partition π a partition Π ⊂ Fθδ [B \ C] such that Var K , Fθδ , F, B < |F (M )| (y,M )∈Π This contradiction proves the lemma Ì ĨƯ Đ 2º Let F be a charge Then the variational measure VM {∅}, F , F, · is σ-finite on a measurable set E ⊂ Rm iff it is σ-finite on each negligible set B ⊂ E which is a Gδ -set in E In particular, absolutely continuous variational measure VM {∅}, F , 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 Prove the sufficiency Consider an arbitrary +∞ It is easy to sequence rn n=1 monotone convergent to zero with r1 ∈ 0, 2m see that bases Frn generated by F satisfy conditions of Lemma Moreover, by the Vitali covering theorem [34, Ch 4, § 3, Theorem 3.1] and by (1) all these 755 FRANCESCO TULONE — YURIJ ZHEREBYOV bases have the Vitali property It follows from (6) that variational measures V {∅}, Frn , F, · are σ-finite on each negligible set B ⊂ E which is a Gδ -set in E (n = 1, 2, 3, ) Then by Lemma variational measures V {∅}, Frn , F, · are σ-finite on E Let Cn = x ∈ E : D Frn |F |(x) < +∞ Since rn -regular derivates DFrn |F | are measurable (see [34, Ch 4, § 4, Theorem 4.2]), sets Cn are also measurable and by Lemma µ(E \ Cn ) = +∞ Let C = n=1 (n = 1, 2, 3, ) Cn All rn -regular derivates D Frn |F | are finite at each point of C Hence, in view of rn → and by (3), all upper derivates D Fr |F | r ∈ (0, r1 ] are finite at each point of C Using an analogue of Ward’s theorem for the basis F ([13, Theorem 3.3]) we deduce that the ordinary derivative FF is finite almost everywhere on C Let D = x ∈ C : |FF (x)| < +∞ and Yk = x ∈ D : |FF (x)| < k ∩ B(0, k) Since the ordinary derivative FF is measurable, the sets D and Yk are measurable too It means that there exist Fσ -sets Pk ⊂ Yk with µ(Yk \ Pk ) = By (3) the r-regular derivative DFr F exists at Fix an arbitrary r ∈ 0, 2m each point x ∈ Pk and DFr F (x) = FF (x) < k Then for each x ∈ Pk there exists δ(x) > such that for each pair (x, M ) ∈ Frδ [{x}], F (M ) (14) By definition, DB |F |(x) = +∞ at each point x ∈ X1 , therefore D B |F | dµ = +∞ (L) (15) X1 Moreover, for any N > and for any positive function δ : X1 → (0, +∞) there +∞ exists a sequence of B-sets Mj (x) j=1 such that for each pair x, Mj (x) ∈ Bδ [{x}] the following inequality is fulfilled F Mj (x) µ Mj (x) 758 > N (16) ABSOLUTELY CONTINUOUS VARIATIONAL MEASURES OF MAWHIN’S TYPE +∞ Since B is a Perron and Vitali basis, the family Mj (x) j=1, x∈X forms a Vitali cover of X1 By the Vitali property there exists at most countable subfamily +∞ M (xk ) k=1 of pairwise disjoint B-sets such that +∞ µ X1 \ M (xk ) = (17) k=1 Pairs xk , M (xk ) we obtain p k=1 form a δ-fine partition anchored in X1 , therefore by (16) p p Var Bδ , F, X1 F M (xk ) >N p µ M (xk ) k=1 M (xk ) = Nµ k=1 k=1 (18) In view of continuity of µ we get Var Bδ , F, X1 (18) +∞ Nµ M (xk ) k=1 +∞ N µ X1 ∩ M (xk ) k=1 +∞ (17) = N µ X1 ∩ +∞ M (xk ) k=1 + µ X1 \ M (xk ) k=1 (14) = N µ(X1 ) > Since N > and the positive function δ(·) are arbitrary, the last inequality implies (19) V B, F, X1 = +∞ Thus, the equalities (15) and (19) prove (12) Now prove (13) Since it is clear for negligible sets we can assume µ(X2 ) > Due to σ-finiteness of µ we can also assume µ(X2 ) < +∞ Thus, < µ(X2 ) < +∞ (20) Consider the sets Dk0 = x ∈ X2 : D B |F |(x) = , Dkn = x ∈ X2 : n−1 2k < D B |F |(x) and the sequence of step functions n fk (x) = k , when n 2k k, n = 1, 2, 3, x ∈ Dkn (n = 0, 1, 2, ) 759 FRANCESCO TULONE — YURIJ ZHEREBYOV +∞ It is clear that, when k is fixed, the sets Dkn are pairwise disjoint, X2 = +∞ n=0 Dkn , the sequence fk k=1 of nonnegative real-valued functions uniformly converges to D B |F |, provided k → +∞, and D B |F |(x) fk (x) < D B |F |(x) + 2k for each x ∈ X2 (21) Due to the measurability of the upper derivate DB |F |, the sets Dkn are measurable Therefore, there exist open sets Gkn ⊃ Dkn such that µ(Gk0 ) < µ(Dk0 ) + and µ(Gkn ) < µ(Dkn ) + n 2n+1 (k, n = 1, 2, 3, ) (22) By definition, there is a gage δkn : Dkn → [0, +∞) such that V B, F, Dkn + Var Bδkn , F, Dkn , 2k+n V B, F, Dkn < +∞ when (23) and Var Bδkn , F, Dkn = +∞, when V B, F, Dkn = +∞ (k = 1, 2, 3, ; n = 0, 1, 2, 3, ) (24) Besides, at each point x ∈ Dkn there exists δkn (x) > such that F (M ) n < k + k+n+1 µ(M ) 2 (k = 1, 2, 3, ; n = 0, 1, 2, ) (25) for each pair (x, M ) ∈ Bδkn [{x}] Define the gage δ as δ(x) = δkn (x), δkn (x), d(x, Rm \ Gkn ) , when x ∈ Dkn (n = 0, 1, 2, 3, ) +∞ There also exists a sequence of B-sets Mknj (x) j=1 associated with the point x ∈ Dkn such that pairs x, Mknj (x) ∈ Bδ [{x}] and F Mknj (x) µ Mknj (x) > n−1 2k (j, k = 1, 2, 3, ; n = 2, 3, ) (26) +∞ Since B is a Perron and Vitali derivation basis, the system Mknj (x) j=1, x∈D kn forms a Vitali cover of Dkn By the Vitali property there exists at most countable +∞ subcover of pairwise disjoint B-sets M (xkni ) i=1 such that +∞ µ Dkn \ M (xkni ) i=1 760 =0 (k = 1, 2, 3, ; n = 2, 3, ) (27) ABSOLUTELY CONTINUOUS VARIATIONAL MEASURES OF MAWHIN’S TYPE The pairs p form a δ-fine partition anchored in Dkn , there- Var Bδ , F, Dkn F M (xkni ) fore p i=1 xkni , M (xkni ) Var Bδkn , F, Dkn Hence, i=1 +∞ Var Bδkn , F, Dkn F M (xkni ) (k = 1, 2, 3, ; n = 2, 3, ) (28) i=1 In view of σ-additivity of the variational measure V B, F, · we get +∞ (L) X2 +∞ +∞ n n−1 µ(Dkn ) fk dµ = µ(Dkn ) = µ(Dkn ) + k k 2 2k n=0 n=2 n=1 +∞ +∞ n−1 µ Dkn \ 2k n=2 M (xkni ) i=1 +∞ + µ Dkn ∩ n−1 µ 2k n=2 = (26) n=2 i=1 +∞ +∞ < M (xkni ) i=1 +∞ (27) +∞ +∞ +∞ M (xkni ) + i=1 +∞ Dkn n=1 µ(X2 ) 2k µ(X2 ) n−1 µ M (xkni ) + k 2k F M (xkni ) + n=2 i=1 (28) +∞ V B, F, Dkn + n=2 +∞ = V B, F, n=2 V B, F, X2 + µ(X2 ) 2k + 2k+n +∞ Dkn (29) µ(X2 ) 2k Var Bδkn , F, Dkn + n=2 (23),(24) +∞ + kµ + n=2 2k+n µ(X2 ) 2k + µ(X2 ) 2k µ(X2 ) + 2k Hence, by (20) and by Fatou’s lemma we obtain (L) X2 DB |F | dµ lim inf (L) k→+∞ fk dµ (29) V B, F, X2 (30) X2 761 FRANCESCO TULONE — YURIJ ZHEREBYOV Let us prove the opposite inequality Let π be an arbitrary δ-fine partition anchored in X2 Then πkn = (x, M ) ∈ π : x ∈ Dkn are δ-fine partitions +∞ anchored in Dkn It is obvious that πkn are pairwise disjoint and π = n=0 πkn (the union is finite, in fact) Let Gkn = M : (x, M ) ∈ πkn Since the B-sets from the partition πkn are nonoverlapping, the following equality holds µ(M ) = µ M (k = 1, 2, 3, ; n = 0, 1, 2, ) M ∈Gkn (x,M )∈πkn By definition of the gage δ we get +∞ F (M ) = F (M ) n=0 (x,M )∈πkn (x,M )∈π (25) +∞ < n=0 (x,M )∈πkn (31) +∞ = n=0 +∞ n=0 (22) < 2k+1 n + k+n+1 µ k 2 M M ∈Gkn n + k+n+1 µ Gkn 2k µ Dk0 + +∞ = n + k+n+1 µ M 2k 2 +∞ + n=1 n + k+n+1 2k µ(Dkn ) n + µ Dkn + k+2 k+n+1 k 2 n=0 n=0 +∞ +∞ n 1 + k n 2n+1 k+n+1 n 2n+1 2 n=1 n=1 +∞ µ(X2 ) + (L) k+n+1 n=0 fk dµ + 2k+2 X2 < µ(X2 ) + (L) 2k D B |F | + = (L) X2 DB |F | dµ + +∞ + 2k+n+1 n=1 +∞ + k+2n+2 n=1 1 1 dµ + k+2 + k+1 + k+2 k 2 2 X2 762 n 2n+1 +∞ + (21) µ Dkn + 2µ(X2 ) + 2k (31) ABSOLUTELY CONTINUOUS VARIATIONAL MEASURES OF MAWHIN’S TYPE Hence, by the definition of the variational measure and by (20) finally we get V B, F, X2 Var Bδ , F, X2 (L) D B |F | dµ (32) X2 Now the equality (13) is a trivial corollary from (30) and (32) The sufficiency follows in a trivial way from the equality V B, F, X = (L) DB |F | dµ = 0, X which holds for each negligible set X ⊂ E The proof is complete Theorem implies a series of corollaries ĨƯĨÐÐ ƯÝ 2º Let B be a derivation BF -basis on a measurable set E ⊂ Rm with the Ward property and satisfying conditions of Theorem Let F be an additive B-set function Then the variational measure V B, F, · is absolutely continuous on E iff |DB F | is the Radon-Nikod´ym derivative of V B, F, · ; i.e., V B, F, X = (L) DB F dµ X for each measurable set X ⊂ E P r o o f It is easy to check that the basis B satisfies conditions of Lemmas and Hence, by these lemmas and by [34, Ch 4, § 4, Theorem 4.2], the upper derivate DB |F | is measurable and finite almost everywhere on E Then by the Ward property the derivative DB F is also measurable and finite almost everywhere on E Let D be the set of points at which DB F is finite Using the inequality F (M ) |F (M )| − |DB F (x)| − DB F (x) µ(M ) µ(M ) we deduce that the function |F | is differentiable at each point of D and DB |F |(x) = |DB F (x)| for x ∈ D Thus, by Theorem and by the latter equality V B, F, X = (L) DB |F | dµ = (L) X DB |F | dµ = (L) X DB F dµ X for each measurable set X ⊂ E ĨƯĨÐÐ ƯÝ 3º Let Br denotes one of the bases BrK H or BrP with bounded sequence P (0 < r < 2m ) and let F be an additive Br -set function If the variational measure V Br , F, · is absolutely continuous on a measurable set coincide E ⊂ Rm , then all variational measures V Br , F, · with r ∈ r, 2m 763 FRANCESCO TULONE — YURIJ ZHEREBYOV P r o o f It is easy to check that the basis Br satisfies conditions of Corollary 2, therefore V Br , F, X = (L) DBr F dµ (33) X By (6) the variational measures V Br , F, · are absolutely continuous, too Thus, (33) holds for all these measures By (3) the derivatives DBr F and DBr F coincide almost everywhere on E Hence, by (33) V Br , F, X = (L) DBr F dµ = V Br , F, X DBr F dµ = (L) X X for each measurable set X ⊂ E Consider now the Radon-Nikod´ ym theorems for Mawhin’s type variational measures Ì ĨƯ Đ 4º Let B be the full basis BK H or a P-adic basis B P with bounded sequence P and let F be an additive B-set function Then the variational measure VM B, F, · is absolutely continuous on a measurable set E ⊂ Rm iff |FB | is the Radon-Nikod´ym derivative of VM B, F, · ; i.e., VM B, F, X = (L) FB dµ X for each measurable set X ⊂ E In addition, all variational measures V Br , F, · r ∈ 0, 2m coincide with VM B, F, · P r o o f In view of (6) the absolute continuity of VM B, F, · implies the abso1 lute continuity of variational measures V Br , F, · for each r ∈ 0, 2m Thus, by Corollary all variational measures V Br , F, · coincide Hence, VM B, F, X = sup V Br , F, X = V Br , F, X (34) r ] The bases for each measurable set X ⊂ E and for each regularity r ∈ (0, 2m Br satisfy conditions of Corollary 2, therefore, analysing its proof, we can deduce that derivatives DBr F are measurable and finite almost everywhere on E Hence, by (3) these derivatives coincide almost everywhere It means that the ordinary derivative FB (x) = DBr F (x) almost everywhere on E 764 r ∈ 0, 2m (35) ABSOLUTELY CONTINUOUS VARIATIONAL MEASURES OF MAWHIN’S TYPE Finally we get VM B, F, X (34) = V Br , F, X (33) (35) DBr F dµ = (L) = (L) X FB dµ X for each measurable set X ⊂ E Theorem was proved in [10, Theorem 3.3] in the case of the full basis B K H Here we have given a new proof of this result and extended it also for P-adic bases Now give a new proof of the analogous result obtained for charges by Z Buczolich and W F Pfeffer (see [13, Proposition 4.2, Corollary 4.8], [15, Theorem 4.6], [32, Proposition 3.2, Theorem 3.6]) Ì ĨƯ Đ 5º Let F be a charge Then the variational measure VM F , F, · VM BV , F, · is absolutely continuous on a measurable set E ⊂ Rm iff |FF | |FBV | is the Radon-Nikod´ym derivative of VM F , F, · VM BV , F, · ; i.e., VM F , F, X = (L) FF dµ X VM BV , F, X = (L) (36) FBV dµ X for each measurable set X ⊂ E In addition, all variational measures V Fr , F, · V BV r , F, · r ∈ 0, 2m , coincide with VM F , F, · VM BV , F, · , provided P r o o f Following the proof of Theorem we get that the ordinary derivative FF is measurable and finite almost everywhere on E By (3) DFr F (x) = FF (x) almost everywhere on E r ∈ 0, 2m (37) As in the proof of Corollary we deduce that D Fr |F |(x) = DFr |F |(x) = DFr F (x) r ∈ 0, 2m almost everywhere on E (38) By Theorem 3, applied to the bases Fr1 , Fr2 and to an arbitrary measurable set X ⊂ E, we get 765 FRANCESCO TULONE — YURIJ ZHEREBYOV V Fr1 , F, X (11) (38) (37) D Fr1 |F | dµ = (L) = (L) X (37) DFr1 F dµ = (L) X (38) X X (11) D Fr2 |F | dµ = V Fr2 , F, X , DFr2 F dµ = (L) = (L) FF dµ X (39) provided < r1 < r2 2m Hence, the variational measures V Fr , F, · coincide By definition of VM F , F, · we finally get VM F , F, X = sup V Fr , F, X = V Fr , F, X (40) r ] Now equality (36) for each measurable set X ⊂ E and for each r ∈ (0, 2m follows directly from (39) and (40) The same statement for the variational measure VM BV , F, · follows from Lemma and [13, Lemma 3.1] The proof is complete REFERENCES [1] BOGACHEV, V I.: Foundations of Measure Theory, Vols 1, Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk, 2003 [2] BONGIORNO, B.: Essential variations In: Measure Theory Oberwolfach 1981 Lecture Notes in Math 945, Springer-Verlag, Berlin, 1981, pp 187–193 [3] BONGIORNO, B.—DI PIAZZA, L.—PREISS, D.: Infinite Variation and Derivatives in Rm , J Math Anal Appl 224 (1998), 22–33 [4] BONGIORNO, B.—DI PIAZZA, L.—SKVORTSOV, V.: A new full descriptive characterization of Denjoy-Perron integral, Real Anal Exchange 21 (1995/96), 656–663 [5] BONGIORNO, B.—DI PIAZZA, L.—SKVORTSOV, V.: The essential variation of a 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dyadic KurzweilHenstock integral by means of variational measure In: Proc of the XXVI Conference of Young Scientists, Faculty of Mechanics and Mathematics, Moscow State University, Vol 1, Moscow State University Press, Moscow, 2004, pp 93–96 (Russian) [49] ZHEREBYOV, YU A.: Variational measures in the theory of the integral Ph.D Dissertation, Moscow State University, Moscow, 2006 (Russian) Received 19 2009 Accepted 28 10 2009 * Department of Mathematics Palermo University Via Archirafi 34 I–90123 Palermo ITALY E-mail : tulone@math.unipa.it ** RUSSIA E-mail : yurazherebyov@gmail.com 768 ... 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 748 ABSOLUTELY CONTINUOUS VARIATIONAL MEASURES OF MAWHIN? ? ?S TYPE of E (for... etc But the case of Mawhin? ? ?s type variational measures still has not been considered In this paper we give similar characterizations for several variational measures of this type (see Theorem and... measures in the metric space (W, d) Unless specified otherwise, by σ-finiteness of variational measures V (B, F, ·) and VM (B, F, ·) we understand their σ-finiteness as Borel measures It follows