On Non-metric Covering Lemmas and Extended Lebesgue-differentiation Theorem

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1 British Journal of Mathematics & Computer Science 8(3): , 205, rticle no.bjmcs ISSN: SCIENCEDOMIN international On Non-metric Covering Lemmas and Extended Lebesgue-differentiation Theorem Mangatiana. Robdera Department of Mathematics, University of Botswana, Private Bag 0022, Gaborone, Botswana. rticle Information DOI: /BJMCS/205/6752 Editor(s): () Paul Bracken, Department of Mathematics, University of Texas-Pan merican Edinburg, TX 78539, US. Reviewers: () nonymous, Turkey. (2) Danyal Soyba, Erciyes University, Turkey. (3) nonymous, Italy. Complete Peer review History: Original Research rticle Received: 2 February 205 ccepted: 2 March 205 Published: 28 March 205 bstract We give a non-metric version of the Besicovitch Covering Lemma and an extension of the Lebesgue-Differentiation Theorem into the setting of the integration theory of vector valued functions. Keywords: Covering lemmas; vector integration; Hardy-Littlewood maximal function; Lebesgue differentiation theorem. 200 Mathematics Subject Classification: 28B05; 28B20; 28C5; 28C20; 46G2 Introduction The Vitali Covering Lemma that has widespread use in analysis essentially states that given a set E R n with Lebesgue measure λ(e) < and a cover of E by balls of arbitrary small Lebesgue measure, one can find almost cover of E by a finite number of pairwise disjoint balls from the given cover. The more elaborate and more powerful Besicovitch Covering Lemma [, 2] is known to work for every locally finite Borel measures. Its setting is a finite-dimensional normed space X. For an arbitrary X, and a family of balls B(a, r a) such that a and sup a r a <, the theorem states that there exists a constant K depending only on the normed space X such that for some *Corresponding author: robdera@yahoo.com

2 Robdera; BJMCS, 8(3), , 205; rticle no.bjmcs m < K, one can find m disjoint subsets i such that for each i, the balls B(a i, r ai ) are pairwise disjoint and m a i B(a, r a) still covers. The Besicovitch Covering Lemma is used to prove the important Lebesgue-differentiation Theorem: Theorem.. Let µ be a locally finite Borel measure on R n. Given x suppµ, f L (R n, µ), then lim fdµ = f(x), µ a.e. r 0 µ(b(x, r)) B(x,r) Here, suppµ is Ω \ U, where U is the largest open set such that µ(u) = 0 and B(x, r) is the ball centered at x of radius r. Such a theorem provides an important tool in many areas of analysis, such as, partiall differential equations, harmonic analysis, probability, integral operators, approximation theory, to name just a few. Obviously, such a theorem has several proofs, and extensions in the literature (see e.g. [3, 4, 5, 6, 7]). The aim of this note is to give a more general, non-metric Besicovitch Covering Lemma that works for any size function (see definition in Section 2) and that will permit to further extend the scope of the Lebesgue Differentiation Theorem to the more general setting of the integral of vector valued functions as introduced in [8] (further developed in [9]). 2 Extended Notion of Integrability In this section, we recall the main points in the definition of the extended notion of integrability as introduced in [8]. size function is set function µ : Σ 2 Ω [0, ] defined on a semiring Σ of subsets of Ω satisfying µ( ) = 0; µ() µ(b) whenever B in Σ (monotonicity) µ( n) n) for every sequence n n in Σ such that n N n Nµ( subadditivity). n N n Σ (countable Σ-subpartition P of a subset 2 Ω is any finite collection {I i; I i, I i Σ, i =, 2..., n} with the following properties that µ(i i) < for all i {,..., n}, I i, I i Σ and I i I j = whenever i j. Σ-subpartition P = {I i : i =,..., n} is said to be tagged if a point t i I i is chosen for each i {,..., n}. We write P := {(I i, t i) : i {,..., n}} if we wish to specify the tagging points. We denote by Π(, Σ) the collection of all tagged Σ-subpartitions of the set. The mesh or the norm of P Π(, Σ) is defined to be P = max{µ(i i) : I i P }. If P, Q Π(, Σ), we say that Q is a refinement of P and we write Q P if Q P and P Q. It is readily seen that the relation is transitive on Π(, Σ), and if P, Q Π(, Σ), then P Q := {I \ J, I J, J \ I : I P, J Q} Π(, Σ), P Q P and P Q Q. Thus the relation has the upper bound property on Π(, Σ). We then infer that the set Π(, Σ) is directed by the binary relation. Let f : Ω V, where V will denote either a real or a complex normed vector space. Given a Σ-subpartition P = {(I i, t i) : i {,..., n}} Π(, Σ), we define the (Σ, µ)-riemann sum of f at P to be the vector f µ(p ) = n µ(ii)f(ti). Thus the function P fµ(p ) is a V -valued net defined on the directed set (Π(, Σ), ). We thereby say that Definition 2.. function f : Ω V is (Σ, µ)-integrable over a set Σ, with (Σ, µ)-integral fdµ if for every ɛ > 0, there exists P0 Π(, Σ), such that for every P Π(, Σ), P P0 we have fdµ f µ(p ) < ɛ. (2.) 22

3 Robdera; BJMCS, 8(3), , 205; rticle no.bjmcs We shall denote by I(, V, Σ, µ) the set of all (Σ, µ)-integrable functions over the set. It should also be noticed that if the set is such that µ() = 0, then for all subpartitions P Π(), f µ(p ) = 0, and thus fdµ = 0. It follows that fdµ = gdµ whenever µ{x : f(x) g(x)} = 0. We write f µ g, if µ{x : f(x) g(x)} = 0. It is readily seen that the relation f µ g is an equivalence relation on I(, V, Σ, µ). We shall then denote by I(, V, Σ, µ) the quotient space I(, V, Σ, µ)/ µ. For p <, we shall denote by I p (Ω, V, Σ, µ) the subspace of I(Ω, V, Σ, µ) consisting of functions f such that the function s f(s) p V is µ-integrable. The space Ip (Ω, V, Σ, µ) shall be normed by f f p = ( Ω g p V dµ) p. The space I (Ω, V, Σ, µ) is defined by I (Ω, V, Σ, µ) = { f F(Ω, V ) : µ-esssup f V < } where F(Ω, V ) denotes the set of all V -valued function defined on Ω. The space I (Ω, V, Σ, µ) will be normed by f f = µ esssup f V. Remark 2.. We notice that: ) the spaces (I p (Ω, V, Σ, µ), p, p <, and ( ) I (Ω, V, Σ, µ), are all Banach spaces whenever V is a Banach space. if µ is the Lebesgue measure, then the Lebesgue function space L p (Ω, V ) is contained in I p (Ω, V, (Σ, µ)). a continuous version of the Dvorestski-Rogers theorem proved in [9, Theorem 6] shows that I (Ω, V, (Σ, µ)) I(Ω, V, (Σ, µ)). in the above definition of the integrability, no notion of measurability is required. the Hölder s inequality has the following generalization: if p, p 2,..., p n, r [, ) satisfy p + p p n = r then for every f i I p i (Ω, R, Σ, µ), i {, 2,..., n}, we have [ ] f f 2 f n r r n [ dµ f i p pi i dµ]. Ω Ω For more detailed exposition and further results on the notion of extended integral see [0]. 3 n Extension of the Besicovitch Covering Lemma In what follows, (Ω, τ) is a topological vector space and Σ τ, and µ : Σ 2 Ω translation invariant size function, that is, [0, ] is a Σ, ω Ω, µ(ω + ) = µ(). It is a well-known fact that a topological vector space has a local base consisting of balanced sets. We shall denote by Nx bal the collection of all balanced neighborhoods of x Ω. We introduce the notion of µ-balls that will take on the role played by metric balls in the standard Besicovitch covering theorem. 222

4 Robdera; BJMCS, 8(3), , 205; rticle no.bjmcs Definition 3.. Let 0 be the zero vector in Ω. We shall call a µ-ball of size r centered at 0, the set of the form N(0, r) = { } N N0 bal, N is open, µ(n) > r. For an arbitrary element ω Ω, a µ-ball of size r centered at ω is defined to be the set N(ω, r) := ω + N(0, r). Clearly, N(ω, r) is balanced, and if r < r, then N(ω, r) N(ω, r ). By translation invariance, we have µ(n(ω, r)) = µ(n(0, r)). More generally, given a subset of Ω, a set of the form N(, r) := + N(0, r) is called a µ-neighborhood of the set. We say that a subset 2 Ω is µ-bounded if there exists r > 0 such that N(0, r). Definition 3.2. We say that the size function µ satisfies the uniform doubling condition if there exists k > 0 such that for every ω Ω and for every r > 0, we have µ(n(ω, r) + N(0, r)) = kµ(n(0, r)). (3.) It is easily verified that the Carathéodory extension of the Lebesgue measure is an example of size function that satisfies the uniform doubling condition. In what follows we shall always assume that µ satisfies the uniform doubling condition. We need some technical lemmas. Lemma 3.. ssume that the size function µ satisfies the uniform doubling condition. Then there exists an integer K 0 such that if ω, ω,..., ω K0 Ω with the properties that N(ω, ) N i(ω i, ) for each i, then some N i(ω i, ) contains ω j for some i j. Proof. Let ω,..., ω m Ω be such that N(ω, ) N(ω i, ) for each i {,..., m}, and no ω i belongs to N(ω j, ) for i j. Then we notice that N(ω i, ) N(ω, ) + N(0, ) for each i. Indeed, if x N(ω i, ), then x ω 0 N(0, ) where ω 0 N(ω, ) N(ω i, ). On the other hand, ω 0 ω N(0, ), and so x ω = (x ω 0) + (ω 0 ω) N(0, ) + N(0, ) which implies that It then follows that x ω + N(0, ) + N(0, ) = N(ω, ) + N(0, ). m N(ω i, ) N(ω, ) + N(0, ). We next claim that N(ω i, ) are disjoint µ-balls, that are obviously contained in m 2k N(ωi, ). Indeed, if N(ω i, ) N(ωj, ), then 2k 2k It follows from (3.) that N(ω i, N(ω i, 2k ) N(ωj, 2k ) N(ωi, 2k ) + N(0, 2k ) (3.2) 2k ) + N(0, 2k ) N(ωi, ) N(ωi, ). (3.3) 2 Combining (3.2) and (3.3) would imply that ω j N(ω i, ). This contradiction proves our claim. It then follows again from (3.) that 223

5 Robdera; BJMCS, 8(3), , 205; rticle no.bjmcs m 2k ( m m ) = µ (N(ω i, /2k)) = µ N(ω i, /2k) ( m ) µ N(ω i, ) µ (N(ω, ) + N(0, )) = kµ(n(0, )) = k. Hence m 2k 2. The proof is complete. With a few obvious technical changes, the following variant of the above lemma can safely be established. Lemma 3.2. There exists an integer K such that if ω, ω,..., ω K Ω with the properties that N(ω, r) N(ω i, r i) for each i, where r i 2 r > 0, then some Ni(ωi, ri) contains ωj for some 3 i j. We now state and prove our first extension of the Besicovitch covering lemma. The proof follows the same transcendental induction argument as in the proof of the corresponding standard metric case (see e.g. []). One simply replaces closed balls with µ-balls. Theorem 3.3. Let F be a collection of µ-balls of size at most R, for some R > 0. Let C be the set of the centers of the µ-balls in F. Then there exists an integer K > 0, such that for each k < K, there exists a countable collection C k F such that C K k= N C k N. Proof. Let R 0 := sup {µ(n) :N F}. Pick N(ω, r) F such that r 9 R0. We consider the integer 0 K given by Lemma 3.2. Define F = {N(ω, r)}, F 2 = F 3 =... = F K = where K = K +. We let T = {N(ω, r) F : ω N(ω, r)} and R = F \ T F F K. Let H be the collection of partitions of F of the form {F,, F K, T, R} with the following properties:. Each F j consists of disjoint µ-balls (from F). 2. If N(ω, r) F j, and if N(ω, r ) F is such that ω N(ω, 9 0 r), then N(ω, r ) T. 3. If N(ω, r) F j, and N(ω, r ) R, then r 9 0 r. 4. If N(ω, r) T, then some N(ω, r ) F F K. The step above shows that H =. We partially order H as follows {F,, F K, T, R} F i F i, i, T T, R R. { F,, F K, T, R } We are done if we show that there exists {F,, F K, T, R} H such that R =. Let {F α,, FK, α T α, R α } α be a totally ordered family in H. Then clearly, { F α,, FK, α T α, } R α α α α α 224

6 Robdera; BJMCS, 8(3), , 205; rticle no.bjmcs is an upper bound. Hence by the Zorn s Lemma, H has a maximal element, say M = {F,, F K, T, R}. We shall show that R =. ssume that R =. Let R := sup {µ(n) :N R}. Pick N(ω, r) R such that r 9 R. Then N(ω, r) is disjoint from all the balls of at least one of the Fi, because 0 otherwise we would have a contradiction with Lemma 3.2. Let k be the smallest element in {,..., K} with such a disjointness property. Consider M := {F,, F k {N(ω, r)}, F k +,..., T, R } where T are those balls in R whose centers are in N(ω, r), and R denotes the reminder. Then M H and M M. This contradicts the maximality of M, and hence finishes our proof. We now introduce the definition of a non-metric Besicovitch covering. Definition 3.3. Let F be a collection of non trivial µ-balls in Ω. We say that F is a µ-fine Besicovitch covering for a set 2 Ω if for every a and every ɛ > 0, there exists a µ-ball N(a, r) F such that r < ɛ. Our next result generalizes the Besicovitch measure-theoretical covering Lemma. Theorem 3.4. Let Ω, µ() <, and let F be a µ-fine Besicovitch covering of. Then there exists a countable subcollection C of F, consisting of disjoint µ-balls such that µ( \ N C N) = 0. (3.4) Proof. We assume that µ() > 0, otherwise the statement is trivial. Since µ() <, we also can assume without loss of generality that the µ-balls elements of F are all of size at most. Then by Lemma 3.3, there exists an integer K > 0, such that for each k < K, there exists a countable collection C k F such that K k= N C k N. Hence there exist k {,..., K}, and disjoint µ-balls in F centered at ω,..., ω L Ω such that It follows that L µ( \ L µ( N(ω i, r i)) µ() K +. N(ω i, r i)) ( K + )µ(). Next, we let 2 = \ L Ni. If µ(2) = 0, the process terminates and the theorem is proven. Otherwise we let F 2 = {N F : center of N in 2, N N(ω i, r i) =, i =,..., L } and we then apply the same argument as above to the pair ( 2, F 2) to obtain disjoints µ-balls centered at ω L +,..., ω L2 Ω such that L 2 µ( \ N(ω i, r i)) ( K + )2 µ(). Repeating the process m times, we will obtain a collection of L m µ-balls centered at ω Lm +,..., ω Lm Ω such that m µ( \ N(ω i, r i)) ( K + )m µ(). (3.5) 225

7 Robdera; BJMCS, 8(3), , 205; rticle no.bjmcs If for some m N, µ( \ m N(ω i, r i)) = 0 the process terminates and the theorem is proven. Otherwise, (3.5) holds for all m N. Letting m, the claim follows. 4 Maximal Function In this section we extend the Lebesgue-Differentiation Theorem to the setting of functions in the space I (Ω, X, µ) where Ω is a topological vector space, X is a normed vector space and µ is a translation invariant size function that satisfies the uniform doubling condition. Definition 4.. Let f I (Ω, X, µ). We define the Hardy-Littlewood Maximal Function M µf by M µf(x) = sup f(ω) dµ(ω). r>0 µ(n(x, r)) N(x,r) Further, and more generally, if ν : Σ X is an additive vector measure of bounded variation, then we define ν(n(x, r)) M µν(x) = sup r>0 µ(n(x, r)). Recall that the variation of an additive vector measure ν : Σ X is defined to be ν = sup ν() π π where the supremum is taken over all Σ-partitions π of Ω. The principal result about maximal function is the following: Proposition 4.. Let µ be a translation invariant size function that satisfies the uniform doubling condition. If ν : Σ X is an additive vector measure and α > 0, then µ ({x Ω : M µν(x) > α}) α ν. (4.) Here and hereforth {M µν > α} is a short for {x Ω : M µνf(x) > α}. Proof. Let = {M µν > α}, and let F consist of all µ-balls N(x, r x) of size at most r x, centered at points x of such that ν(n(x, r)) > α. µ(n(x, r)) If =, there is nothing to prove. If, the hypotheses of Theorem 3.4 are satisfied, and it follows that there exists a family of disjoint µ-balls {N(x i, r i) F : i N} such that µ(\ N(xi, ri)) = 0. Hence µ() µ(n i) ν(n(x, r)) α α ν. The proof is complete. by Since every f I (Ω, X, µ) naturally defines an additive set function of bounded variation given ν() = fdµ, the following result is a particular case of the above proposition. 226

8 Robdera; BJMCS, 8(3), , 205; rticle no.bjmcs Proposition 4.2. Let µ be a translation invariant size function that satisfies the uniform doubling condition. If f I (Ω, X, µ) and α > 0, then We now prove the Lebesgue-Differentiation Theorem. µ ({x Ω : M µf(x) > α}) α f. (4.2) Theorem 4.. Let Ω be a locally compact topological vector space. Let µ : Σ 2 Ω [0, ] is a finitely additive size function. If f I (Ω, X, µ), then lim fdµ = f(x), µ-a.e. r 0 µ(n(0, r)) N(0,r) Proof. By local compactness of Ω, it is quickly seen that f I (Ω, X, µ) if and only if Kf I (Ω, X, µ) for every K compact subset of Ω. We then may assume without loss of generality that Ω is compact. Given ɛ > 0, let g be continuous on Ω such that f g < ɛ. Since we have for every x Ω, we have for every α > 0, { E α := = lim r 0 x Ω : lim sup r 0 { x Ω : lim sup r 0 gdµ = g(x) µ(n(0, r)) N(0,r) µ(n(0, r) µ(n(0, r)) N(0,r) N(0,r) } fdµ f(x) > α } (f g)dµ (f g)(x) > α. We are done if we show that µ(e α) = 0. We notice that lim sup (f g)dµ (f g)(x) < Mµ(f g) + f g. r 0 µ(n(0, r)) N(0,r) It follows that E α { x Ω : M µ(f g) > α } { x Ω : f(x) g(x) > α }. 2 2 By Proposition 4. and the Tchebytchev s inequality, it follows that µ(e α) 4 α f g < 4ɛ α for all ɛ > 0. Hence µ(e α) = 0. The proof is complete. 5 Conclusion We have given a treatment of the Lebesgue-Differentiation Theorem in the setting of the integration theory of vector valued functions. The treatment leans heavily of the machinery developped in the first half of the paper which provides a non-metric version of the Besicovitch covering lemma. The author believes that the interest in such generalizations lies not only in the fact that we have more general results, but also in the light they shed on the classical situation. Competing Interests uthor declares that no competing interests exist. 227

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