FUNDAMENTAL THEOREM OF CALCULUS FOR HENSTOCK -KURZWEIL INTEGRAL

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1 Bulletin of te Maratwada Matematical Society Vol. 14, No. 1, June 2013, Pages FUNDAMENTAL THEOREM OF CALCULUS FOR HENSTOCK -KURZWEIL INTEGRAL Anil Pedgaonkar, Institute of Science, Mumbai , Abstract An introduction to Henstock-kurzweil integral is given. Te integral is Super Lebesgue and also includes improper Riemann integrals. Every derivative is integrable and tis result is proved. We obtain a mean value teorem for vector valued mappings. A generalization of te notion of Euclidean space, namely a metric semigroup is discussed. It provides an abstract framework wic includes te integral for mappings assuming fuzzy real values as well as Aumann integral for set valued mappings introduced by Nobel Laureate Aumann. An advantage of tis approac is tat one need not know wat is a fuzzy real number but one just needs to know tat fuzzy real numbers form a metric semigroup. A version of fundamental teorem of calculus as well as mean value teorem is proved for mappings assuming values in a Metric semigroup. Tus we obtain anoter approac to define Aumann integral for set valued mappings, were we do not ave to consider integrable selectors. It is also to be noted tat to define Lebesgue integral we need te notion of norm and in case of metric semigroups we do not ave te notion of norm as well as a linear functional. Apart from outer measure, or Daniell lattice, Henstock - Kurzeweil integral offers yet anoter metod to construct Lebesgue measure. 1 1 INTRODUCTION, BASIC NOTATION AND CONVENTIONS We introduce te Henstock Kurzweil integral for mappings defined on te real line in Section 2, and derive te most satisfactory virsion of Fundamental Teorem of Calculus tat every derivative is integrable. We give an example of a function wic is Henstock-Kurzweil integrable but not Riemann integrable as well as not Lebesgue integrable. Since te definition of integral is based on Riemann sums and does not involve te notion of absolute value or norm, it is possible to constructively define Henstock- Kurzweil integral for mappings wose codomain may not ave a natural definition of 1 AMS subject classification, 28B15 Keywords : Aumann Integral, Henstock -Kurzweil Integral, Fuzzy real numbers, Fundamental Teorem of Calculus, Mean Value Maratwada Matematical Society, Aurangabad, India, ISSN

2 72 Anil Pedgaonkar norm but only a topology defined on it. Henstock -Kurzweil integral for mappings assuming values in a metric semigroup is introduced in te tird section. Te integral includes te integral for fuzzy real valued mappings and Aumann integral for set valued mappings. We prove te fundamental teorem of Calculus for mappings assuming values in a in a metric semigroup. We point out tat we ave anoter approac to Aumann integral of set valued mappings wic does not require te use of integrable selectors. We follow [6] and use te term mapping. Te term function is reserved for mappings wose codomain is te set of real numbers. As in [6], x denotes te norm of a vector x in a Euclidean space. ℵ and R respectively denote te set of Natural numbers and te set of real numbers. We stipulate tat 0. = 0. 2 HENSTOCK-KURZWEIL INTEGRAL ON A COMPACT INTERVAL Consider mappings defined on a compact interval [a, b] on te real line, assuming values in a Euclidean space. Definition 2.1 By tagged partition of [a, b], we mean division of [a, b] into subintervals J k, = [y k, x k ], 1 k n, y k < x k,, y 1 = a, x n = b and tags t k J k for eac k. Let te lengt of subinterval J k be denoted by J k = x k y k. By a Riemann sum associated wit a tagged partition P, we mean te sum n f(t k ). J k.. If f is unbounded at a tag t k, or is not defined at t k, we simply omit te corresponding term from te sum. Te sum is denoted by S(P, f). Definition 2.2 A gauge δ on R is strictly positive function δ, defined on R. Given a gauge δ, a tagged partition P = {(t k, J k )} of an interval is said to be δ fine, if for eac k, J k δ(t k ). Teorem 2.1 (Cousin s Lemma) δ fine partition exists for any gauge δ. Proof: Suppose te contrary. We divide [a, b] on two subintervals [a, c] and [c, b], were c is midpoint of [a, b]. At least one of te subintervals does not ave δ fine partition. We denote it by I 1. We bisect I 1. At least one of te subintervals does not posses a δ fine partition, and denote it by I 2. Continuing we obtain a sequence of nested subintervals I n wose diameter tends to 0. By Cantor s teorem tere exists a unique point x in te intersection. Since diameter tends to 0 tere exists some n large enoug so tat I n δ(x). Tus we arrive at a contradiction, and te proof is complete. Remark 2.1 Henstock-Kurzweil, integral is defined in a way similar to Riemann integral. Te only difference is tat wile any point is allowed as a tag point in te Riemann integral and only constant gauges are considered, in Henstock-Kurzweil integral we use a gauge function δ wic may vary at eac point and allow only δ fine tagged partitions. So te irregularities of te mapping can be better controlled by a subinterval of a smaller size containing te tag. Since a smaller family of partitions is used, more mappings are integrable.

3 Fundamental Teorem of Calculus for Henstock Definition 2.3 Ẇe say a mapping f (possibly unbounded) is Henstock-Kurzweil integrable on [a, b], if tere exists a vector I wit te property tat, given ϵ > 0, tere exists a gauge δ, suc tat, for any δ fine partition of [a, b], S(P, f) I < ϵ. We say tat I is te Integral of f and te integral is denoted it by I(f) or by [3] for more details. Teorem 2.2 Henstock-Kurzweil integral is well defined. b a f(t)dt. Refer [2] or Proof: Suppose I is anoter value of integral. Hence given ϵ > 0, tere is a gauge δ 1 and tere is a gauge δ 2 suc tat for any δ 1 fine partition P 1 and any δ 2 fine partition P 2 of [a,b], we ave S(P 1, F ) I < ϵ/2 and S(P 2, F ) I < ϵ/2. Select a gauge δ = δ 1 δ 2,,defined as, δ(t) = min{δ 1 (t), δ 2 (t) : t R}. Ten for a, δ fine partition P, I S(P, f) < ϵ/2 and I S(P, f) < ϵ/2. Hence I I < ϵ. Since ϵ is arbitrary, I = I, and te proof is complete. Teorem 2.3 (Fundamental Teorem of Calculus) If a mapping F is continuous on [a, b], differentiable on (a, b) wit f as it s derivative, ten f is integrable on [a, b]. Furter, I(f) = b a f(x)dx = F (b) F (a). Proof: To prove te teorem we select a gauge using straddle lemma given below. Lemma 2.1 (Straddle Lemma) If F is differentiable in a neigborood of a point t, ten for eac ϵ > 0, tere exists a neigborood of t say J t, suc tat wenever x > t > y are in J t, we ave F (x) F (y) f(t)(x y) ϵ(x y). To prove tis lemma, we observe tat f is differentiable at t, and ence, given ϵ > 0, tere exists a neigborood of t say J t, suc tat for x > t > y, x, y J t,, F (x) F (y) f(t)(x t) ϵ(x y). Similarly, F (t) F (y) f(t)(t y) ϵ(t y). Lemma follows from te triangle inequality and, te order inequality x > t > y. Te lemma is illustrated in te diagram. Remark 2.2 It is important tat x and y straddle t, tat is x and y occur on different sides of t. Te lemma asserts tat te slope of te cord joining te points wit ordinates x and y is a good approximation to te slope of te tangent at te point were ordinate is t. To continue te proof of te teorem, let ϵ > 0,. We select te desired gauge δ using straddle lemma. For eac point t (a, b) we associate an interval J t as given by straddle lemma. Let J t = te lengt of J t. Let δ t = min{ J t, t a, b t}, for eac t (a, b). For te point t = a, since f is continuous at a, we ave an interval [a, r] suc tat for t [a, r], f(a) (x a) < ϵ we let δ(a) to be min{r a, s a} ten we let J a = [a, a+δ(a)].

4 74 Anil Pedgaonkar In a similar fasion we select δ(b) > 0 and J b = [b δ(b), b]. We consider a δ fine tagged partition P. Note tat a and b are forced to tags for any δ fine taggged partition P, by construction of δ. Te tags of P are t 1 = a, t 1... t n 1, t n = b, and te associated subinterval wit eac t k is, J k = [y k, x k ], k = 1, 2,... n. F (b) F (a) = = n n 1 k=2 n f(t k )(x k y k ). S(P, f) F (b) F (a) = ϵ(x k y k ) + ϵ + ϵ ϵ(2 + b a) and te proof is complete. F (x k ) f(y k ).S(P, f) n f(t k )(x k y k ) [(F (x k )) F (y k )] Remark 2.3 Every Lebesgue integrable mapping is Henstock-Kurzweil integrable. (See [1], [2]) We now illustrate te use of Fundamental Teorem of Calculus wit an example. We use te result tat every derivative is Henstock-Kurzweil integrable. However we sow tat te function is neiter Riemann nor Lebesgue integrable. Example 2.1 Define a function F on [0, 1] by F (0) = 0, oterwise F (x) = x 2 cos(π/x 2 ). Te derivative of F is f, given by, f(0) = 0, f(x) = 2x cos(π/x 2 ) + (2π/x) sin(π/x 2 ), if x 0. Since f is unbounded, f is not Riemann integrable, but f is Henstock integrable, by te fundamental teorem of calculus. Integral of f is 1. Moreover f is not Lebesgue integrable eiter. To see tis let 0 < a < b < 1, ten f is Riemann integrable on [a, b] as f is continuous and value of integral is b 2 cos π/b 2 a 2 cos π/a 2 1. Set a k = 2k and 2 b k = 4k+1.Ten integral of f on [a k, b k ] is 1 2k. Since te intervals [a k, b k ] are pair wise disjoint and infinite in number, we ave, integral of f on [0, 1] is Σ1/(2k) =. So f is not absolutely integrable and ence f is not Lebesgue integrable. Teorem 2.4 Mean value Teorem for Vector valued mappings: Let F be a map, continuous on [a, b], and differentiable in (a, b), ten F (b) F (a) = 1 0 F (a + θ)dθ, were = b a. Proof 2.1 By cain rule F is a mapping of θ and for t = a + θ, derivative of F wit respect to t is derivative of F wit respect to θ multiplied by. Te proof is immediate by an application of fundamental teorem of Calculus, noting tat is constant. Remarks: 1) Te mean value teorem is a crucial tool to establis many results in differential calculus. We ave obtained a version of mean value teorem as an equality in integral form. Since very derivative is integrable, additional assumptions on te derivative are unnecessary.

5 Fundamental Teorem of Calculus for Henstock ) Taylor s teorem for vector valued mappings wit integral form of remainder can be obtained by using integration by parts. We need not assume continuity of derivative of n+1 t order, if we use Henstock-Kurzweil integral and tus it is an indispensable tool for differential calculus of vector valued mappings or differential Calculus in Banac spaces. We refer te reader to [6], were differential calculus is carried out in Euclidean spaces, but results are not presented in complete generality as Henstock-Kurzweil integral is not used. 3) Since definition of te integral does not use te notion of norm, te integral can be defined, for mappings assuming values in a topological vector space or a topological semigroup, or set valued mappings in contrast to te Lebesgue integral. To illustrate te ideas more clea1ly we discuss a classical example in integration teory. Example 2.2 Diricilet function is Henstock- kurzweil integrable Consider f : [0, 1] R, f(x) = 1 wen x is rational and 0 wen x is irrational. Since te set of rational numbers is countable we let r j, j = 1..., be an enumeration of te set of rational numbers in [0, 1]. Let ϵ > 0 be given. Define a gauge δ as, δ(t) = ϵ/2, if t is irrational. If t is rational ten t = r j for some j and ten we define δ(t) = ϵ/(4.2 1 j ). For irrational tags t, te contribution to Riemann sum is 0. If t is a rational tag in I k, ten I k 2 1 j, (ϵ/4). Since no point can occur as tag in more tan 2 subintervals, contribution to te Riemann sum from rational tags is 2.Σ2 1 j (ϵ/4) = ϵ. f is integrable and I(f) = 0. Remarks: 1) we can define Henstock - Kurzweil intregral on unbounded intervals as well. 2) Te usual linearity properties of te integral and te additivity over te interval old. We now present a treatment of te integral wen mappings assume values in a metric semigroup. Tis serves to model te mappings assuming fuzzy real values as well as Aumann integral of set valued mappings used in Economics. 3 METRIC SEMIGROUPS A metric semigroup is a generalization of te notion of an Euclidean space. Integral for metric semigroup valued mappings is introduced in [4]. An important example of a metric semigroup is furnised by te set of fuzzy real numbers L(R). We refer te reader to [7] for an introduction to fuzzy numbers. However we prefer to treat general metric semigroups. We need not know about fuzzy real numbers. Definition 3.1 A metric semigroup is a structure (Y, d, +,.), were i) (Y, +) is a commutative semigroup and ii) (Y, d) is a complete metric space, satisfying te following properties iii) Te scalar multiplication is left distributive. λ(x + y) = λx + λy, for λ R, and x, y Y.

6 76 Anil Pedgaonkar iv) (λ + µ).x = λx + µx, for x Y and λ, µ R + {λ, µ 0} v) d(a + b, c + d) d(a, c) + d(b, d), a, b, c, d Y. vi) d(λa, λb) λ d(a, b) for λ R, a, b Y. vii) 1.x = x, for all x in X. Te metric semigroup is called invariant if d(a + c, b + c) = d(a, b), for a, b, c Y. As stated above one example is furnised by te set of fuzzy real numbers L[R]. We now give an example of a metric semigroup oter tan L(R). Example 3.1 Consider te following families of subsets of R. C = te family of nonempty closed subsets of R, K = nonempty compact subsets of R, K c = family of nonempty closed convex subsets of R. We define (Minkowaski) addition and scalar multiplication by A + B = {a + b : a A, b B} and λa = {λa : a A}. Te standard Hausdroff metric dh, for a pair of sets is defined as follows. Define Excess of A over B = e(a, B) = sup{d(a, B) : a A}, Define excess of B over A = e(b, A) = sup{d(a, b) : b B}. We define Hausdroff metric as d H (A, B) = max{e(a, B), e(b, A)}. Hereafter we use te symbol d for d H as it is te only metric under consideration on family of sets. It is sown [5, pp 9-10], tat under tis metric, eac of te above families is a metric space and tus furnises an example of metric semigroup. We illustrate te Minkowaski addition and scalar multiplication by an example. Example 3.2 Consider A = [0, 1]. Hence ( 1).A = [ 1, 0].A + ( 1).A {O}, but A + ( I).A = [ 1, 1]. Definition 3.2 (of Integral) Let f be a mapping defined on an interval B, assuming values in a metric semigroup Y. Given a tagged partition P of B, te Riemann sum is defined by S(P, f) = n J k.f(t k ). We define f to be integrable wit te integral denoted by I(f, B) or simply by I(f), if given any ϵ > 0 tere exists a gauge δ, suc tat for any δ fine tagged partition of B, d(s(p, f), I(f)) < ϵ. Remark 3.1 We can sow tat most of te usual properties of te integral old including linearity. We now extend te fundamental Teorem of Calculus to te setting of Metric semigruops. Definition 3.3 (Hakukara Difference) Let a and b be two elements in a metric semigroup. We define te Hakukara difference a b as a b = c, if b + c = a. We note tat Hakukara difference may not exist, for eac pair a, b Ṗroposition 3.1 Hakukara difference of any two elements, wen it exists, is unique.

7 Fundamental Teorem of Calculus for Henstock Proof Let a b = c, and a b = d. Hence a = b + c = b + d,, d(c, d) d(b + c, b + d) = d(a, a) = 0. Hence c = d. Remark 3.2 Hakukara differenc a b may not be confused wit a + ( 1)b. If 0 exists in te Hakukara semigroup ten a a = 0, even toug a + ( 1)a may not be equal to 0. Example 3.3 Consider te sets in Example 3.3 [ 1, 1] [0, 1] = [ 1, 0]. Example 3.4 Te Hakukara difference {0} [0, 1] does not exist, as no translate of [0, 1] can ever belong to {0}. Proposition 3.2 b + (a b) = (a b) + b = a, provided te Hakukara difference a b is defined. Proof: Let a b = c. ten b + c = a by definition. Hence b + a b = b + c = a. Te oter part follows by commutative property. Proposition 3.3 (a + x) (b + y) = (a b) + (x y), provided all differences exist. Proof: Let a b = c, x y = z. Hence a = c + b, x = z + y. We get, a + x = (c + z) + (b + y). Proposition 3.4 (a b) + (b c) = a c. Proof: let a b = x, b c = y. Ten a = b + c, b = c + y. Hence a = c + x + y. Hence a c = x + y. Remark 3.3 ( Limit and continuity) Since (Y, d) is a metric space we can define limits and continuity of any mapping f, assuming values in Y. Definition 3.4 (Hakukara Derivative) We say a mapping F defined on (a, b) and assuming values in a metric semigroup, is Hakukara differentiable at a point, t, in (a, b), if tere exists an element S in Y suc tat, lim d 0 ( ) {F (t + ) F (t)}, S = lim 0 d ( {F (t) F (t )} ), S = 0, (only for te points for wic Hakukara difference exists are considered wile computing te limit). We denote te value, S, by F (t). If F is differentiable at every point t in (a, b) ten we say F is differentiable in (a, b) and tis gives a new mapping from (a, b) to Y, called as derivative of F and is denoted by F Remark 3.4 Te Hakukara difference F (t ) F (t) does not always exist even for a simple linear mapping F (t) = t.a. So we ave taken te difference in reverse order in te definition. We illustrate te penomenon by an example. Consider, for example, A = [0, 1] as in te Example 3.3. Let t = 1, F (t) = t.a, F (1) = A, F (1 ) = (1 ).[0, 1] = [0, 1 ], wen 0 < < 1. No translate of A = [0, 1] can give [0, 1 ] wen > 0.

8 78 Anil Pedgaonkar Teorem 3.1 Hakukara derivative is unique, wenever it exists. Proof: Let if possible ( s and r be ) be ( two values for ) derivative of F at t. Ten d(s, r) d F (t+) F (t), s + d F (t+) F (t), r. Taking limits as 0, d(s, r) = 0, since te distance between two points s and r is constant. Hence s = r. Since our main objective is to prove fundamental teorem of Calculus we merely state te following result for completeness toug it is not used in te proof. Teorem 3.2 If f and g are differentiable at a point t ten: i)f + g is differentiable at t and (f + g) (t) = f (t) + g (t). ii)λf is differentiable and (λf) = λf,for any λ in R. Teorem 3.3 (Fundamental Teorem of Calculus) F is continuous on [a, b], taking values in a metric semigroup, Hakukara differentiable on [a, b] wit f as it s derivative. Ten f is integrable on [a, b] and F (b) = F (a) + b a f(x)dx. To prove te teorem, we select a gauge using straddle lemma. Lemma 3.1 (Straddle Lemma): If F is Hakukara differentiable in a neigborood of a point t, ten for eac ϵ > 0, tere exists a neigborood of t say J t suc tat wenever, x > t > y are in J t, we ave, d(f (x) F (y), f(t).(x y)) 2ϵ(x y). Proof As f is differentiable at t, given ϵ > 0, tere exists a neigborood of t say J t suc tat for ( ) ( ) {F (x) F (t)} {F (t) F (y)} x > t > y, x, y J t, d, f(t) < ϵ, d, f(t) < ϵ. x t t y Since x t and t y are greater tan 0, we can use property (vi) of metric semigroups. d(f (x) F (t), (x t).f (t)) < ϵ.(x t) and d(f (t) F (y), (t y).f (t)) < ϵ.(t y). We use property (v) and Proposition So conclusion of straddle lemma follows. Proof of te Teorem For a given ϵ > 0, select a gauge using straddle lemma. For eac point t in [a, b] we associate an interval J t by straddle lemma. Let δ(t) = J t = te lengt of J t. Consider a δ fine tagged partition P of [a, b], wit t 1, t 2, t 3,..., t n as tag points, wit associated subinterval for eac t k as J k = (y k, x k ). Now d(s(p, f), [F (b) F (a)]) = d( n f(t k )(x k y k ), n [F (x k ) F (y k )]) We ave used proposition 3.12 in te transformation of F (b) F (a) to te sum of terms containing Hakukara differences, eac term being, [(F (x k ) F (y k )]. d( n f(t k ).(x k y k ), [F (x k ) F (y k )]) (by property (v)) n 2ϵ(x k y k ) = 2ϵ(b a). And te proof can be completed. Remark 3.5 It is easily possible to modify te above proof so tat differentiability of F is assumed only in (a, b). In tis case we can define f(a) = f(b) = 0, witout altering te integrability of f as te value of te integral. We use continuity of F at a and b.

9 Fundamental Teorem of Calculus for Henstock Teorem 3.4 Mean Value teorem for mappings taking values in a metric semigroup. Let f be a map, f is continuous on te [a, b], and differentiable in [a, b], ten f(b) = f(a) f (a + θ)dθ, were = b a. Proof: By cain rule f is a differentiable function of θ and for t = a + θ, derivative of f wit respect to t is derivative of f respect to θ multiplied by. Te proof is immediate by an application of fundamental teorem of Calculus, and noting tat is constant. Remark 3.6 Te standard proof of cain rule works, as [0, 1] is mapped onto [a, b] and since te integral is defined using Riemann sums, constant can be pulled out of integral sign. Corollary 3.1 If te derivative of a mapping F is zero in an interval ten F is constant. Proof: We apply te mean value teorem. Since integral of te zero mapping is zero (Riemann sum being zero) F (b) = F (a) for any two points a, b in te interval. Remark 3.7 Wen we consider metric semigroup, to be te collection of compact convex subsets we get Aumann integral as discussed in [1]. But in Henstock-Kurzweil integration we do not ave to consider integrable selectors, and te integral is defined directly using te Riemann sum. We illustrate tis by giving some examples. Te second example is adapted from [5], were te differentiability of te mapping is discussed. Example 3.5 Let A = [0, 1]. Let Y be te collection of compact intervals in R. Let F : [0, 1] Y be given by F (t) = t.a, were A = [0, 1]. We find tat F (t + ) F (t) =.A, for eac t [0, 1] and > 0. Hence F (t) = A, for eac t [0, 1]. So if we define f(t) = A, for eac t [0, 1], ten 1 0 f(t).dt = F (1) F (0) = A {0} = A. Example 3.6 Let B be te closed unit ball in te plane. Let F be defined on [0, 2π] F (t+) F (t) as,f (t) = (2+sin t).b. If t [0, π/2], > 0, = 1/.[2 sin /2. cos(t+/2).b] F (t) F (t ) F (s+) F (s) and as 0 te Limit is cos(t.b). Also = (by writing t as s) = 1/.[2 sin /2. cos (s + /2).B. As 0, te limit is cos t.b. So we ave, F (t) = f(t) = cos t.b. Hence π/2 0 f(t)dt = F (π/2) F (0) = 3.B 2.B = B. However as mentioned in [5], te Hakukara derivative does not exist in [π/2, 3π/2] as in order tat Hakukara difference, be defined, te translate of F (t) must be contained in F (t + ). So te diameter F (t), must be a nondecreasing function. Tis is not true, wen t [π/2, 3π/2]. CONCLUSIONS We ave obtained generalized fundamental teorem of calculus for Hakukara semigroups wic in particular covers Aumann integral and integral for mappings assuming fuzzy real numbers as values. We ave sown tat integrable selectors are not necessary and Aumann integral can be defined directly.

10 80 Anil Pedgaonkar References [1] Aumann R.,J Integrals of set valued functions, J. Mat Analysis and Applications, Vol , [2] Bartle R. G., A Modern Teory of integration, GSM, Vol 22; AMS Publication. [3] Bartle R. G. and Serbert D, Introduction to Real analysis - Tird edition, Jon Wiley and Sons, New York [4] Boccuto A., Candeloro D, Sambucini A., Integrals for Metric semigroup valued functions defined on unbounded intervals, Pan American Mat. Journal,Vol 17, 2004, [5] Diamond Pil and Kloeden P., Metric spaces of fuzzy sets valued Function Teory and applications, World Scientific, 1994, 1 32, [6] Lang Serge, Analysis I, Addison - Wesley Publising company, New York, [7] Nguyen H. Walker E., A First course in fuzzy logic, Tird edition, Capman and Hall/CRC Press, 2005, 1-14,