Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 682960, 28 pagesdoi:10.1155/2012/682960

Research Article

Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces

Maciej Ciesielski¹ and Anna Kamińska²

¹Institute of Mathematics, Poznań University of Technology, Piotrowo 3a, 60-965 Poznań, Poland²Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA

Received 10 October 2011; Accepted 20 January 2012

Academic Editor: Lech Maligranda

Copyright © 2012 Maciej Ciesielski and Anna Kamińska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spaces and in particular on Lorentz spaces for any and a nonnegative locally integrable weight function , where is a maximal function of the decreasing rearrangement for any measurable function on , with . The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages of , where is an order continuous r.i. quasi-Banach space. The second type of LDT establishes conditions for pointwise convergence of the best or extended best constant approximants of or , , respectively. In the last section it is shown that the extended best constant approximant operator assumes a unique constant value for any function , .

1. Introduction

The present paper is devoted to investigation of maximal inequalities and Lebesgue’s type differentiation theorems for best local approximations in r.i. quasi-Banach spaces and Lorentz spaces for . In 1910, Henry Lebesgue has proved one of the most famous differentiation theorem, which establishes a convergence of an integral average of any locally integrable function on the ball to this function , that is, for a.a. , In fact, Lebesgue’s integral average coincides with a best constant approximant on the space [1]. The Lebesgue Differentiation Theorem (LDT) can be proved as a consequence of the weak maximal inequality for the Hardy-Littlewood maximal function where and [2]. The interesting exploration of LDT was initiated by Stein in [3], who introduced the maximal functions on , associated with integral average, and applied it to obtain differentiation theorem in the notation of the norm in for . In the spirit of this idea many authors developed new techniques of recovering functions in quasi-Banach function spaces. The first results in this subject were obtained by Bastero et al. [4] in 1999, who have investigated Hardy-Littlewood maximal functions and weak maximal inequalities in rearrangement invariant quasi-Banach function spaces. The next paper was published by Mazzone and Cuenya in 2001 [1], about generalizations of the classical Lebesgue differentiation theorem for the best local approximation by constants over balls in for . They evaluated maximal inequalities for the maximal function related to best constant approximants and proved convergence theorem for best constant approximants. In 2008 [5] Levis et al. extended the best constant approximant operator from Orlicz-Lorentz spaces to the spaces and showed monotonicity of the extended operator. In view of this result, in 2009 [6] Levis established maximal inequalities for the maximal function associated with the best constant approximation and proved Lebesgue’s type differentiation theorem for best constant approximants and for integral averages expressed in terms of the modular corresponding to these spaces. Recently, the authors have characterized properties of an expansion of the best constant approximant operator from Lorentz spaces to the spaces . The present paper is a continuation of the previous results and devoted to investigation of maximal inequalities and Lebesgue’s type differentiation theorems for local approximation in r.i. quasi-Banach space and in particular in .

The paper consists of three sections and is organized as follows.

In the preliminaries, Section 2, we establish some basic notations and definitions and also recall some auxiliary results, which will be used later.

Sections 3 and 4 consist of the main results of the paper.

We start Section 3 proving measurability of the maximal function for , that corresponds to the quasi-norm average of , in r.i. quasi-Banach function spaces . Next we establish two types of generalization of LDT in r.i. quasi-Banach function spaces and in . In both types of LDT we employ the assumption of upper and lower -estimates of . The first main result, in the spirit of Stein [3], has been proved for any order continuous r.i. quasi-Banach function space. The statement is expressed in terms of quasi-norm averages. In order to show it we first prove the inequality for maximal function , which corresponds to a quasi-norm average of . In the same spirit we also provide some conditions when the LDT does not hold in or in . Next we continue our discussion with another type of LDT. In order to complete the second main result in this section we characterize conditions for which Lorentz space satisfies a lower (resp., an upper) -estimate, where is the fundamental function of . In view of this characterization we investigate pointwise convergence of the best constant approximants to as whenever and , as well as the convergence of the extended best constant approximants for any and . We also present examples showing that this assumption is fulfilled by a large class of the spaces . Finally, we investigate relations between maximal functions and the -functional of Banach couple in the spirit of the inequalities stated in [4]. We finish Section 3 with an example showing that is not equivalent to the -functional of the pair .

It is well known that the extension of the best constant approximant operator from to , or from to , is a set valued function [1, 7]. Contrary to this in Theorem 4.5 we prove that the extended best constant approximant operator assumes a unique value for any and . To show the uniqueness we need to consider strict monotonicity of the right-hand Gâteaux derivative of the norm in at in the direction for any and .

2. Preliminaries

Let and be the set of real and natural numbers, respectively. For any denote . Let and be the Lebesgue measure on . We denote by the space of all extended real-valued -measurable and finite functions a.e. on . Denote the outer measure on by , the support of by , and the restriction of to the set by . By a simple (resp., step) function we mean a measurable function with a finite measure support, which attains a finite number of values (resp., a finite number of values on a finite number of disjoint intervals). The distribution function of a function is given by for all . Two functions are called equimeasurable, if for all . We define the decreasing rearrangement for any by , . For given we denote the maximal function of by . It is well known that and is decreasing and subadditive, that is, for any . For the properties of , , and see [2, 8]. A subspace equipped with a quasinorm is called a quasinormed function space, if the following conditions are satisfied.(1)If , , and a.e., then and .(2)There exists a strictly positive .

If is complete, then it is said to be a quasi-Banach function space. We say that a quasi-Banach function space is rearrangement invariant (r.i. for short), if whenever and with , then and (see [2]). Throughout the paper we use the notation , which means that the expressions and are equivalent; that is, is bounded from both sides. Let and be a nonnegative weight function. Lorentz space is a subspace of such that Given a measurable set by we denote the set of restricted to and satisfying the above inequality. Unless we say otherwise, throughout the paper we assume that belongs to the class (in short ), whenever it satisfies the following conditions: for all if and for all otherwise. These two conditions guarantee that . We also assume that Under these assumptions is a rearrangement invariant (r.i. for short) quasi-Banach function space such that it has the Fatou property and the order continuous norm. Letting and , , the space will be denoted by .

Unless we say otherwise, throughout this paper we assume that is the fundamental function of defined as , , and . It is easy to show that the fundamental function is strictly increasing and continuous on , and for . For more details about the properties of see [9].

Recall that for given , classical Lorentz space is a subspace of such that In case when satisfies the -condition, that is for all and some , as well as , the space is a separable r.i. order continuous quasi-Banach function space [9]. The space is a r.i. Banach function space, whenever the weight is decreasing and [10]. Since , we have the natural inclusion . Moreover, if and only if satisfies condition , ( for short) which means that there is such that for all we have [11–13].

Let and be -finite measure spaces. A map from into is said to be a measure-preserving transformation, if whenever is a -measurable subset of , the set is a -measurable subset of and . For given subsets such that , there exists a measure-preserving transformation [14, Theorem 17, page 410].

Definition 2.1 (see [15]). Let . Denote for all .

In 1970, Ryff proved in [16] that is a measure-preserving transformation for any and a.e. on . In 1993, Carothers et al. established in [15] that is a measure preserving transformation from onto such that a.e. on for any with for all and any . Notice that for any with for any and , if for every we have that and it is the unique measure-preserving transformation up to measure zero satisfying a.e. on .

Definition 2.2. Let and let , be measure-preserving transformations given by Definition 2.1. Denote

Definition 2.3. Let and with . Denote for any and .

Let () be a real normed space. Denote by (resp., ) the closed unit ball (resp., the unit sphere) of . Assume that is a subset of and is an element of . An element is called best approximant to from if A nonempty subset of is a set of uniqueness if for any element there is no more than one element satisfying (2.8). The set is a set of existence if for every element there is at least one element for which condition (2.8) holds. The set is a Chebyshev set if for every element there exists exactly one element satisfying (2.8), that is, if is both a set of uniqueness and a set of existence (for more details see [17]). Let be a collection of all subsets of . A set value map is said to be best approximant operator, if it assumes for any a set of all best approximant elements to from , that is, In case when is a norm function space and is a family of constant functions, then is called best constant approximant operator, and each element is called best constant approximant to from . Let with and . It is well known that the set is convex, compact, and a set of existence for all [18, 19]. Let’s recall some characterizations of best constant approximants over Lorentz spaces .

Theorem 2.4 (see [20, Theorem 7.5]). Let and let . Then is the best constant approximant of if and only if

Corollary 2.5 (see [20, Corollary 7.3]). For any and positive, we have that is Chebyshev set for any ; that is, there is an unique best constant approximant to .

Recently, in [7] it has been developed an existence of extension of the best constant approximant operators from Lorentz space to , if , and from to , if and . Now we recall definition of the extended operator on , if , and on , if .

Definition 2.6 (see [7]). Let with and let if , and if . Assume that if , and if . Denote Then the extended best constant approximant operator is given by

In fact, any is called an extended best constant approximant of . Notice that in view of Theorem 2.4, if for , then any is a classical best constant approximant of .

Definition 2.7 (see [4]). Let , be a r.i. quasi-Banach function space, and let be an increasing bijection. is said to satisfy an upper (resp., a lower) -estimate for , if there exists such that for all and with pairwise disjoint supports we have respectively

In the case when and , this definition covers the notions of the upper (resp., lower) -estimate [21].

Let and be a quasi-Banach function space. We denote by the -convexification of equipped with the quasinorm . Now we recall the definition of the maximal function for any r.i. quasi-Banach function space, that plays a crucial role in process of generalization of Lebesgue’s Differentiation Theorem in .

Definition 2.8. Let , be a r.i. quasi-Banach function space. For any we denote for all and . The maximal function is given by for any , where .

We finish the preliminaries with the following proposition needed further. It is a generalization of the well-known result, which in particular can be found in [22] for special case when on . The proof of the proposition is quite standard and is provided for the sake of completeness.

Proposition 2.9. Let , , and let be an increasing continuous function. If or for all , then

Proof. Notice first if and there exists such that , then , and the conclusion follows.Now assume that is a nonnegative simple function, where for any , , and whenever . Then for any , where and for . We claim that By monotonicity and continuity of we obtain . On the other hand we have . Therefore , which implies condition (2.18). Now we will show For we have , and so . On the other hand, for every , and consequently, , which provides (2.19). Both (2.18) and (2.19) show (2.17) for any nonnegative simple function .Now suppose that is a measurable function and for any . Then, by standard arguments of existing a sequence of nonnegative simple functions such that and as we can show that and conclude the proof.

In fact, Proposition 2.9 describes the largest family of increasing and continuous functions , for which (2.17) is satisfied. Indeed, let , , and for any . Then , for all and for any . Clearly, is increasing and continuous and also . Therefore, and , which implies that condition (2.17) does not hold.

3. Lebesgue’s Differentiation Theorems

The intention of this section is to establish generalizations of LDT in r.i. quasi-Banach function spaces in terms of the formulas expressed by quasinorm averages. We also focus on convergence of the best and the extended best constant approximant of to , which is another type of LDT. First we introduce the notion of the differentiation property for a quasi-Banach function space .

Definition 3.1. Let be a quasi-Banach function space on . We say that has the Lebesgue differentiation property (LDP), whenever for any and for a.a. we have

Observe that letting be a quasi-Banach function space on with LDP, by the Aoki-Rolewicz theorem [23] there exist and an equivalent -norm to such that for any and for a.a. we get If is a normed space, then the quasinorm can be replaced by .

In the next proposition we establish measurability of the maximal function .

Proposition 3.2. Let and let be a r.i. order continuous quasi-Banach function space. If , then the maximal function is measurable on .

Proof. Let and . We first observe that is continuous on . In fact, for any , , and by order continuity of we obtain that and . Now by Fatou’s property of we have Hence we have and thus is measurable.

Remark 3.3. If satisfies condition, then we obtain measurability of the maximal function for any , analogously as in case of the maximal function for any when and .

In view of Theorem  1 in [4], we investigate the so-called weak inequality for the maximal function whenever is a r.i. quasi-Banach function space.

Theorem 3.4. Let . If a r.i. quasi-Banach function space satisfies a lower -estimate for , then there exists such that for all and we have

Proof. Assume that . Denote Clearly, by Proposition 3.2 we get that is measurable for all . Letting , there exists such that and Let and denote . Since , we get . Hence, by regularity of the Lebesgue measure there is a compact set such that . By the fact that a collection is an open covering of the set and by the Vitali covering lemma [2, Lemma 3.2], there exists a pairwise disjoint finite collection such that . Therefore, by and by (3.7) we get Hence, by assumption that satisfies a lower -estimate for , there is such that for any we get Since is arbitrary, we obtain , which finishes the proof.

In the next theorem we present Lebesgue’s differentiation property in the space .

Theorem 3.5. Let and let be a r.i. order continuous quasi-Banach function space . If satisfies a lower -estimate for , then has LDP, that is, for all and for a.a. . If in addition is normable, then

Proof. Observe first that the set of step functions with supports of finite measure is dense in . The proof of this observation is standard, by density of the simple functions, which is equivalent to order continuity of and regularity of the Lebesgue measure on (cf. [8]).Define an operator by for any and . Assume that and and also , . Let be a characteristic function of an open interval . Notice that for a.a. there exist such that for all we have either or and consequently . Therefore, for a.a. and for any . Hence for a.a. . Observe that the above equation can be proved analogously for any step function with support of finite measure. Let be a constant in the triangle inequality of the quasinorm . Thus for a.a. . Clearly , whence for a.a. . Now replacing by we get for a.a. . Define for any . By Proposition 3.2, we have that is -measurable. Recall [2, 8] that for any . Thus in view of (3.17) we obtain for , Now since satisfies the triangle inequality with constant , we get for every . Observe that for any , , we have . Thus, by Proposition 2.9 we have Furthermore, by Theorem 3.4 there exists such that Therefore, for any step function and for all , Hence we have for all . So for a.a. , which shows the first formula.The second formula results from the first one since is a norm in .

Now we characterize the lower and upper -estimate of on , for . Clearly in this case satisfies condition. Thus and . Hence by Theorems 3 and  7 in [9] and by Hölder’s inequality we obtain the following corollary.

Corollary 3.6. Let , and, . (i)If , then satisfies a lower -estimate for the functional , where , and an upper -estimate for , where .(ii)If , then satisfies an upper -estimate for the functional , where , and a lower -estimate for the functional , where .

The immediate consequence of Theorem 3.5 is the next result, which describes under what conditions has LDP. The second part follows from the previous corollary.

Proposition 3.7. Let and . If the Lorentz space satisfies a lower -estimate for , then has LDP. Consequently, for and , Lorentz spaces on have LDP.

We will show next which spaces do not have LDP with respect to . Notice that the ratio can be reduced easily to the quotient for any , where . In view of this fact, we state the following theorem.

Theorem 3.8. Let be a r.i. quasi-Banach function space. (i) Let . If order continuous satisfies a lower -estimate for , then for any and for a.a. we have (ii) Let . Then for any and for a.a. we have

Proof. Let be the operator on given by , .(i) Let be a step function with a finite measure support. Notice that for a.a. and for small enough we have Therefore and thus for a.a. . Denoting , , by Proposition 3.2, is -measurable. Now by Theorem 3.4 there exists such that for all , which shows that and thus a.e.(ii) Let a.e., where are step functions. Therefore, for a.a. there is such that and for small enough we have Thus, by the assumption that we get for a.a. , and the proof is finished.

The next corollary follows directly from Theorem 3.8.

Corollary 3.9. (i) Let , . If Lorentz space satisfies a lower -estimate for , then for any and for a.a. we have (ii) Let , . Then for any and for a.a. we have

The next result needed for further applications states conditions, which guarantee that satisfies a lower -estimate for , where . We omit the proof of the following proposition.

Proposition 3.10. Assume that , and Lorentz space satisfies a lower -estimate. If is concave, then satisfies a lower -estimate for .

The following example shows that Theorem 3.8 is not an empty statement in case of .

Example 3.11. Let , , and for all . Then for any and for a.a we have

Proof. Notice that for any , Consequently, for any . Since is increasing, satisfies condition. Moreover, we have for any , which concludes that is increasing. Hence, by Theorem 3.3 [24] we obtain that satisfies a lower 2-estimate. Now we claim that is concave. By simple calculations we get the second derivative on , We observe that Therefore, for all , which implies concavity of . Hence, by Proposition 3.10 we obtain that satisfies a lower -estimate for . Finally, in view of Corollary 3.9 we finish the proof.

The last part of this section is devoted to a pointwise convergence of the best and the extended best constant approximant of to , that is, another type of LDT.

The first result is a corollary of Proposition 3.7. In fact, let , be such that . Assume that for . By definition of the best constant approximant we get Now applying Proposition 3.7 we get that as . Therefore we get the following theorem.

Theorem 3.12. Let and . If the space satisfies a lower -estimate for , then for every we have for a.a. , where is a best constant approximant of .

In order to prove the next approximation theorem let us first establish an inequality related to extended best constant approximants of any function for .

Lemma 3.13. Let , , , and let , . Then for any extended best constant approximant of and for a.a. we have

Proof. Let and . Since , by subadditivity of the maximal function and the power function for we get We will finish the proof under assumption that . In the other case the proof is similar. Since , by Corollary 2.11 [7] and by Hardy-Littlewood inequality we obtain Moreover, by assumption that we have for any . Consequently, by the fact that we obtain Hence, by conditions (3.43) and (3.44) we finish the proof.

Corollary 3.14. Let and . Assume that and are the fundamental functions of and , respectively. If one of the following conditions (i)there exists such that for all ,(ii), holds, then .

Proof. Notice that for all , Suppose now that there is such that for all . Therefore, for any , and so .Now assume that . Then there exists such that for all we have . Consequently, for any , and the proof is completed.

Theorem 3.15. Let , , and , . Assume that and are the fundamental functions of and , respectively. If Lorentz space satisfies a lower -estimate for and , then for a.a. , where is an extended best constant approximant of .

Proof. Let and , . By Lemma 3.13 for any we get Since , there is such that for all we have . Consequently, for any and . Hence, by Proposition 3.7 and by assumption that satisfies a lower -estimate for we obtain that as for a.a. .

Now we present a specific family of Lorentz spaces for which Theorems 3.12 and 3.15 are fulfilled.

Corollary 3.16. Let and . If one of the following conditions (i) and ,(ii) and , , is satisfied, then for a.a. we have where is the best constant approximant of and the extended best constant approximant of .

Proof. Suppose that condition (i) is fulfilled. Immediately, by Theorem 3.12 and Corollary 3.6 we complete the first case. Now assume that the condition (ii) is satisfied. Define for . Since , we have that for all , which implies that . Consequently, by Definition 2.6 there is the extended best constant approximant operator on . Moreover, by Corollary 3.14 the fundamental functions and of the spaces and respectively, satisfy . Hence, by Corollary 3.6 we get that satisfies a lower -estimate for . Finally, by Theorem 3.15 we obtain that , an extended best constant approximant of , converges to as for a.a. .

Observe that the -functional of the couple can be expressed equivalently by for any and [25]. The next result on inequalities between maximal function and the -functional of follows immediately as a consequence of Theorems 1 and  2 in [4], Corollary 3.6, and Remark 3.3.

Corollary 3.17. Let , , and . (i)If , then there exists such that for all and we have (ii)If , then there exists such that for all and we get (iii)If , then there is such that for any and we obtain the inequality in the condition (i).

The decreasing rearrangement of the maximal function and -functional of are not equivalent; that is the opposite inequalities to the ones in Corollary 3.17 do not hold. It is similar as in spaces , , as we see in the next example [4].

Example 3.18. Let . There exists a function such that for all ,

Proof. Since satisfies condition, there is such that for all , and suitable . Hence . Therefore, by Remark 3.3 we conclude that for all . Now, by Theorem  3 in [4] there exists such that and for all , which implies that and for any .

Remark 3.19. Example 3.18 can be expanded to the case when . Indeed if and then we have and . As well as (·) on . This can be obtained by simple observation that the weight function satisfies condition for any , and then and as sets with equivalent norms.

4. Uniqueness of the Extended Best Constant APProximant on

In this section we prove uniqueness of the expansion of the best constant approximant for any , if and . Namely, we show that the extended best constant approximant operator becomes the point value operator. Throughout this section we assume that is a set of positive and finite measure. Let the truncation of any function be defined as if , and if .

Lemma 4.1. Let and . Then for all .

Proof. Since , we get that for every . Let . Then for any . Moreover, and for any . Therefore, for all , which concludes the proof.

We omit the standard proof of the next lemma.

Lemma 4.2. Let and . Then for a.a. there exists such that for all we have the following. (i).(ii).(iii).

Now we discuss a convergence of a sequence of functions to the function as for all and , whenever .

Theorem 4.3. Let and . Then for all ,

Proof. Denote . Notice that there exists such that and for all we have . Since is a measure-preserving transformation, by Definition 2.3 we get We claim that for a.a. there exists such that for all we have Let and . If , then by Lemma 4.2 there exists such that for any we obtain If , then again by Lemma 4.2 for we have Hence we obtain our claim. It follows for any . Moreover for all , and consequently by triangle inequality for maximal function we obtain for all . Since is the measure-preserving transformation for all , by the fact that the power function for is subadditive with a constant , we have for every and . Combining now (4.8), (4.9), and (4.10) we get In view of (4.4) we complete the proof.

The following characterization of the function is an essential fact in the proof of the main result.

Proposition 4.4. Let , be a positive weight function and let . Then the function is strictly decreasing with respect to .

Proof. First we will show that for any , there exist and , such that for all , Suppose for a contrary that there exist such that for all and for a.a. we have Since and are continuous functions with respect to and is right-continuous with respect to , we get that the above inequality is fulfilled for all and for any . By convexity of the function and by Proposition 4.2 [20] we get that is increasing with respect to for any , which implies that for any and for all . Hence for all and . Pick up . Denote . Notice that for all and , which yields that for all . Consequently, for any . Moreover, by subadditivity of the maximal function and by convexity of the power function for we get for all . Therefore, which implies that for any . Hence for a.a. and for any . By Corollary 9 [8, page 65] we obtain that the functions and for all are of constant sign almost everywhere, that is, for a.a. and have a common system of sets such that and for all and for all . We claim that there exist and , , such that we have either for all , or for all , Suppose that the claim does not hold. Then for any and we get Thus, by the condition (4.24) we obtain for every and . Since for any and for any , we get that for any and for a.a. . Hence, by the fact (4.23) we get a contradiction. Now we will consider two cases.Case 1. Assume that there exist and , such that for all . Thus and by conditions (4.23) and (4.24) we obtain for all . Since , we have for any . The function is strictly increasing on and . In fact, let and . By conditions (4.23) and (4.24) we conclude that for any . Let and . Then, by condition (4.32) we get for every and for all . Repeating calculations in condition (4.32) with instead of , in view of (4.23) and (4.24) we have for all and . Since the functions and are continuous and decreasing with respect to , there exists a compact interval such that for all . Define for any . Since is positive and continuous on , we get . Consequently, there is such that for all . Hence, by condition (4.33) we get for every and . Therefore, by (4.34) we obtain for any and . Now according to Proposition 4.2 [20] we conclude for all , which contradicts condition (4.15) and finishes the first case.Case 2. Suppose that there exist and with such that for any . Analogously as in the previous case we obtain that for all . We claim that the function is strictly decreasing for any . Let and . The conditions (4.23) and (4.24) imply for any . Therefore, for any , and for all we have Repeating calculation in condition (4.43) with instead of , in view of (4.23) and (4.24) we also get for all . Similarly as in the previous case there exists such that for any and . Consequently, by Proposition 4.2 [20] we get for all , which gives us a contradiction with (4.15) and completes the proof of inequality (4.12).Let now and . Denote and . Clearly . By (4.12) there exist and , such that for all . Hence, by (4.14) and by Proposition  2.4 in [7] we get and the proof is done.

Now we establish the main theorem of this section.

Theorem 4.5. Let and be a positive weight function. Then the extended best constant approximant operator assumes an unique value, that is, for any ,

Proof. Suppose that . Then there exist such that . By Theorem 2.9 [7] and by Proposition 4.4 we obtain Let be a sequence of truncations of . By Theorem 4.3 there exists such that for all we have as well as Choose . By Lemma 4.1, for all . Now by conditions (4.52) and (4.53) and by Theorem 2.4 we get that for all . Finally, by Corollary 2.5 we obtain that is unique for , which implies a contradiction and finishes the proof.

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