1. HK integral is invented by Henstock and kurzevill.Every derivative is integrable.The integral is based on Riemann sums. Riemann integration uses partitions whose mesh is smaller than  ( mesh is the size of the largest subinterval) in HK integral we use gauges. The gauges control the size of the blocks of partition. We consider mappings into a Banach space Y. function is a mapping whose codomain is set of real numbers

2. The integral is a natural tool for mappings with values in a Banach space and differential Calculus in Banach spaces. mean value theorem and Taylor’s theorem for vector maps can be proved without imposing any restrictions on nth derivative smaller blocks are used when the mapping is oscillating rapidly in nbd of a point. Gauges can force specific points to be tags, when unusual behavior takes place at these points. numbers.

3. Fundamental Theorem of Calculus (FTC) Gauge : By a gauge  we mean a mapping  which associates to each point x in R* a closed interval (Cell ) containing x in the interior. The interval is bounded unless x is . Fundamental Thm : F is continuous on [a, b] & differentiable on (a, b) with f = F. Then f is integrable on and integral is F(b)– F(a).

4. Proof of FTC uses Straddle lemma stated below If F is differentiable at a point t then for   > 0, ∃ a  nbd of t say J t s.t |F(x) –F(y) – f(t)(x-y)|  . (x - y), whenever x > t > y belong to J t. Proof: |F(x)- F(t) – f(t) (x - t)|   (x- t) & similarly other inequality Lemma follows by triangle inequality and, the order x> t> y.It is important that x and y straddle t that is occur on different sides of t.

5. The lemma asserts that the slope of the chord joining the points with ordinates x and y is a good approximation to the slope of the tangent at the point where ordinate is t. It enables to select a gauge. In metric semigroups we use hakukara derivatives.

6. Proof of FTC using Telescoping sums For a given  > 0 we select a gauge using straddle lemma. For each point t in [a b] we associate an interval J t. let δ(t) = J t |S(P,f) - (F(b) – F(a)| = |  f(t j )(x i – x i-1 ) – [ (F(x i ) – F(x i-1 )]|   (x i – x i-1 ) =  (b –a ). We have extended FTC to mappings assuming values in metric semigroup. This covers fuzzy valued mappings and Aumann integral of set valued maps

7. A function F on [0 1] by F(0) =0, otherwise F (x) = x 2 Cos (  /x 2 ) The derivative is f(x) f(0) = 0, f(x) = 2x Cos(  /x 2 )+ 2  /x. Sin(  /x 2 ),x  0 Since f is unbounded f is not R integrable f is not lebesgue integrable either. But f is HK integrable & value is -1 by FTC Let 0 <a< b<1.f is R integrable on [a b] & integral is b 2 Cos  /b 2 - a 2 Cos  /a 2.. Set a k = 1 /  (2k) and b k =  [ 2 / (4k +1)] then integral of f on [a k b k ] is 1/ 2k. Since the intervals [a k, b k ] are pair wise disjoint & infinite in no. the integral of f on [0 1] is   1 / 2k = . So f is not lebesgue integrable as it is not absolutely integrable.

8. There are no improper integrals in Henstock theory. Drichilet integral 0   (sin x /x)dx exists as HK integral. monotone convergence theorem and dominated convergence theorem hold for HK integrals. An important tool is henstock’s lemma we define a map as Lebesgue ntegrable if both f and |f| are HK integrable. Fourier integral theorem can be extended & applications to integral transforms

9. Taylors Thm for vector valued maps Theorem [51 ] Taylors’ formula : Let f be a mapping defined over an interval [a, b] assuming values in a Banach space, f, and its derivatives up to n th order, are continuous on [a, b] and f (n+1) exists except at a countable set of pts in (a, b), then the mapping f (n+1) (t).(b-a) n is integrable and, f(b) = f (a) +f (1) (a). (b - a) + f (2) (a). +….+f (n) (a) + R n, where R n =

10. Riemann approximation

11. Lebesgue construction

12. Henstock-Kurzweil construction

13. We let R* be R  {  }  {-  }. By extended Euclidean space X of dimension N, we mean Cartesian product of N copies of  *. A point x = ( x 1, x 2,.., x N ) in X is called as point at infinity if at least one coordinate x k, k =1, 2.. N is ∞ or -∞. By a cell in X =  * N, we mean a Cartesian product of N closed intervals, possibly unbounded. A cell is unbounded, if it contains a point at infinity.

14. Integration apparatus/Integrable manifold We have formulated a general framework called as integration apparatus or integrable manifold on which we can carry integration. It covers euclidean spaces as well locally compact T2 spaces,Polonized spaces, Lebesgue Stieltjes integral are also covered by theory It also covers integration of mappings assuming values in a metric semigroup which includes fuzzy valued mappings as well as Aumann integral of set valued mappings

15. Integrable manifold We start a ground set X. Augment it with a set O possibly ( empty set) to form a set X. In the general set up we have the notion of a semilattice of subsets of X denoted by ℬ. Members of ℬ are called cells. ℬ is closed under intersection of sets and X is a cell..

16. The semilatice is called complete when there is a closure operator * defined on cells with the property that If B  D then B*  D*. In the case of Euclidean spaces B* = B that is each cell is a closed cell. We have premeasure defined on ℬ which is positive & additive (defined later) but can take values 0 and . For any cell J, | J |, denotes measure of J, which may be finite or infinite

17. Measured semillatice The measure of a bounded cell is finite. A cell B is unbounded, then B* contains a point from O called as point at infinity and,measure is  We have degenerate cells whose measure is 0, and a cell contained in a degnerate cell is degenerate cell. Typically points in R* are degenerate cells. Points, lines and line segments in plane are degenerate cells.

18. Mutually separated cells:Two cells J and K are called, mutually separated if J  K is a degenerate cell not contained in O (set of pts at infinity). Two nondegeneratee cells B, D are called adjacent if tyhey are separated but not disjoint.For a nondegerate cell B not equal to X,a pt x is called inside B is it does not belong to a cell adjacent to B. The set of inside points is called B . X  = X \O.

19. Partition of a cell : Let B be a cell. By a partition P of B we mean a finite collection P of mutually separated cells given by, P = { B k  ℬ : k =1,2… n. }, such that The union of cells is B The cells B k occurring in the partition are termed as blocks of the partition The measure is called additive if |B| is sum of measures of blocks whenever |B| is finite. We deal with positive additive measure only.

20. Tagged Partition of a cell B:. By a tagged partition P of a cell B, we mean a finite collection P = {(x k,J k ): x k  J* k., k =1, 2.. n}, where the collection { J k } is a partition of B So each block J k is attached with a pt x k, called as a tag.Further x k  J* k. A set is called Negligible if for  > 0,it is covered by a countable collection of mutually separated cells such that sum of their measures is < 

21. Gauges A gauge  is a cell valued mapping defined on X, such that  (x) is a nondegenerate cell containing x, with the property that x is inside  (x),Finite intersection of gauges is a gauge. Given two gauges  and  (    ) (x) =  (x)   (x),  x in X.

22.  Fine tagged Partitions A tagged partition P = { (x k, J k }, k =1, 2.. n} is called  Fine if for each block, J k   (x k ) and J k.is abounded cell when x k is not a pt of O In forming Riemann sum we consider only  Fine partitions and only such tags. So set of partitions is smaller more mappings are integrable.A gauge is called finer than a gauge if  (x)   (x), for all x in X.

26. Axioms AC) Cousin’s axiom : For any cell B, given a gauge , ∃ a  fine tagged partition of B. AD) Division Axiom : If D is a cell contained in a cell B then there exists a partition P of B, such that D is a block in P. AZ) Degeneracy Axiom : Any mapping f, which is zero everywhere, except on a degenerate cell Z is integrable on any cell B, and the value of integral is zero AS) Separation axiom: Given any cell B, there exists a gauge  such that, for all x  B*,  (x) ∩ B* = .

27. Integral and integrable mapping A mapping f is defined to be 0 at a pt at infinity & f is redefined as 0 at pts where f is unbounded.( on a negligible set) so, that f is defined on B.Riemann sum S(P,f) =  f (x j ).|J j |. A map f defined on B* is said to be integrable with integral as I, if for each  > 0,  ∃ a gauge  s.t. | S(P, f) – I | < ,   fine tagged partitions P of B.

28. Integral is Well defined Cousins axiom ensures that the definition is not vacuous Let if possible I & V are two values of the integral.. So given ε > 0, ∃ gauges  1,  2 such that, | S(P, f) – I | < ε/2,   1 fine tagged partitions P,&| S(Q, f) – V | < ε/2   2 fine tagged partition Q of B. Consider the gauge  =  1 ∩  2 finer than both  1,  2. If T is a  fine tagged partition then T is both  1 fine as well as  2 fine. | I – V | ≤ | I – S(T, f) | + | S(T, f ) – V| < ε/2 + ε/2 = ε. So I = V.

29. Mean value theorem Generalized First mean value theorem. g be a nonnegative function defined and integrable over a nondegenerate cell B. Let f be a function defined over a subset A of B* such that that B*\A is a null set and the Extension of f obtained by setting f(x) = 0, for x  B* \A is integrable on B. f has intermediate property on A. The product f.g be Henstock-Kurzweil integrable over B. Then  a point c  A s.t, I(B, f.g) = f(c).I(B, g)

30. Theorem : Given a gauge  there exists a  fine partition of R n Proof: Enough to show for a bounded cell B as in R, = [ m, ] for large M. Suppose there is no  fine partition. Bisect each side & consider all resulting 2 N blocks.At least one of the block say K,does not have any  fine partition. We sudivide the block. Continuing we get a decreasing sequence of cells K k such that none have  fine partition.. Since diameters of K k  0, ∃ an m s.t. x is in K m &  K m   (x). This is a contradiction J m

31. Henstock’s lemma Let f be an integrable mapping over a cell B, with value of integral as F(B), and let the indefinite integral as a set mapping be denoted by F. For given ε > 0, let  be a gauge such that, for any  fine partition P, of B, | S (P, f ) - I | < ε.Then, for any δ fine tagged partial (sub) partition Q, | S(Q, f) - ∑ F(E k )| ≤ ε, where E k ’s are blocks in Q. | S(Q, f) - S(Q, F)| =| S(Q, f – F)| ≤ ε. ii) For real valued functions f, ∑ |f(x k ).| E k | - F( E k )| ≤ 2ε,E k s are blocks of Q

32. Bipartitions enable to prove cousins lemma holds for abstract products and prove Fubini theorem. Bipartitions

33. Net partitions

34. Riemann sum convergence theorem Let B be a cell Let {f n } be a sequence integrable mappings defined over B*, converging to a mapping f almost everywhere on B*. Then, f is integrable over B with I(B, f) = lim I(B, f n ) if and only if, for given ε > 0, ∃ a positive integer n ε such that, for  n ≥ n ε, ∃ a gauge  n such that, for a  n fine tagged partition P of B, d ( S[P, f n ], S[P, f ] ) < ε.

35. Equiintegrability A family H of mappings defined over B*, is said to be equiintegrable if for each ε > 0, there exists a gauge  such that, d (S[P, f], I(B, f)) < ε, whenever P is any  fine tagged partition, and f is any mapping in H. Let{ f n : n  } be equiintegrable sequence of mappings defined over B*, converging to a mapping f pointwise on B*. Then, f is integrable over B and I(B, f)=.Lim (I, f n )

36. Usual Convergence thorems MCT : If {f n } is a nondecreasing sequence of HK integrable functions converging to f, and the sequence of integrals I(f n ) is converging to I, then f is integrable with integral as I. fubini tonnel theorems and Fatous lemma holds verbatiam DCT :If {f n } is a sequence of integrable mappings converging pointwise almost everywhere to a map f and | f n | < g, where g is integrable then f is integrable and I(f n )  I(f).

37. Integral definition of differential form Definition Differential n-1 form on R n :A differential n-1 form ω on  n, is given by ω(x) = dx 1  dx 2  …  dx k *  …  dx n. where * denotes that the corresponding term dx k is omitted. The coordinate mappings f k determine ω completely. It is possible to extend thre idea of next concept and detrivative to abstract set up.

38. Integral definition of dw : Let X be an arbitrary fixed cube in  n and ω be an n-1 form on  n. We say dω is derivative of ω on X if given ε > 0, there exists a gauge  (x) such that, for all x in whenever c is a cube ( of arbitrary orientation ) contained in the cell  (x) | ∫ ∂c w - dω(x). | c | | ≤ ε. | c |. (  c is the boundary of c.) Remark :By a slight abuse of notation we abuse the n -form dω with its single coordinate mapping dω = dω (x). dx 1  dx 2  ….dx n.

39. Induction of measure Integrable Set [32]: A subset A of X is said to be integrable if the characteristic function 1 A is integrable over X. For an integrable set A, we define the measure of A as I (1 A ) A set is called measurable if it is integrable or its intersection with each bounded cell is integrable. Measure of a non integrable but measurable set is set as .

40. Theorem Measure is complete Any subset of a set of measure zero has measure zero. Pf : Supose A has measure zero A is an integrable set with I(1 A ) = 0. Let E be a subset of A. Then 1 E ≤ 1 A. For any partition P, We have |S(P,1 E )| = S(P, 1 E ) ≤ S(P, 1 A ) = | S(P, 1 A ) |. Since 1 A is integrable given ε > 0, we can select a gauge δ such that | S( P, 1 A ) | ≤ ε. Since ε is arbitrary we have that 1 E integrable with value of integral as 0..

41. Dirichilet function f, Hk integrable with f:[0,1]  R,f(x) = 1, x is rational & 0 otherwise Let  > 0 be given. Define a gauge, as  (t) =  /2, if t is irrational,. Since the set of rationals is countable we let r j, j =1…  be an enumeraration of rationals in [0 1].if t is rational then t = r j for some j, define  (t) = (  / 4 ). (1/ 2 j ).Contribution to Riemann sum is 0, From irrational tag points If t is a rational tag pt belonging to block I k, | I k | <= 2 1- j (  /4). No tag can occur in more than 2 subintervals. So contribution from rational tags is  2  2 1-j(  /4) < . So I(f) =0.

42. Illustrations of gauges Example i) Consider the gauge  defined by the following gauge function.  (x) = x/2, if 0 < x ≤ ½  (x) = ½. ( 1 - x), ½ < x < 1.  (-∞) =  (∞) = 100  (x) = 1/5, for any other x. Any tagged partition of [0, 1] must have 0 and 1 as tags

43. Illustrations of gauges Let  be a gauge function defined as,  (x) = x/2, for 0 < x ≤ 1,  (x) = 1/100 for any other x   *. Then any  fine partition of [0, 1] must have 0 as a tag. Suppose we wish to integrate f(x) = 1/root(x) over [0, 1]. F is not defined at 0 and is unbounded near 0. If we use a gauge like , which forces 0 as a tag then the first term in Riemann sum is zero.

44. Alternatively we are ignoring the first term f(0). J 1, but the length of J 1 is less than 1/100 and can be made arbitrarily small by selecting smaller value of  (0). So gauges enable us to control the irregularities of a mapping. Example Consider an interval [a, b] and let c  ( a, b)., and let  1 and  2 be two gauge functions defined on  *.

45. Define a gauge function  defined as,  (x) =  1 (x),if x  [a, c),  (x) =  2 (x), if x  (c, b],  (c) = min {  1 (c),  2 (c) },  (x) is set, arbitrarily at all other points. Any  fine partition P of [a, b] can be written as P 1  P 2, where P 1 is a  1 fine partition of[a, c] and P 2 is a  2 fine partition of [c, b], and c is a tag of two adjacent subintervals. So P is anchored on c. If P 1 is a  1 fine partition of [a, c] and P 2 is a  2 fine tagged partition of [c, b],then P 1  P 2 may not be a  fine partition of [a, b].

46. the gauge function  defined as,  (x) = min {  1 (x), | c -x | if x  [a, c),  (c) = min {  1 (c),  2 (c) },  (x) = min {  2 (x), | c -x |, if x  (c, b].  (x) is set, arbitrarily at all other points. Any  fine partition P of [a, b] must have c as a tag for any subinterval containing c. So every  fine partition P of So every  fine tagged partition P of [a, b] gives rise to two tagged partitions P 1 and P 2, respectively of [a, c], [c, b]. P 1 is  1 fine, P 2 is  2 fine. S(P,f) = S(P 1, f) + S(P 2, f).

47. Alexiewicz norm KH(B, Y) the space of Henstock-Kurzweil integrable mapings is a normed linear space under the Alexiewicz norm defined as || f || kh = sup{ | I (D, f) |: D is a cell contained in B.} It is not complete. Let f n be a mapping on [0, ∞] defined as f n [x] = n, when x in [0, 1/n] and 0 otherwise. Clearly each f n is integrable with integral 1. As n  ∞, f n approaches the generalized function Dirac delta fn.

48. Thank you all ! References : Aumann R. J, Integrals of set valued functions, Journal of. Mathematical Analysis & Applications, Vol 12, 1965, pp 1-12. Anil Pedgaonkar, Henstock integral an abstract approach. ICM 2010, abstracts,pp 226, Anil Pedgaonkar, Fundamental Theorem of Calculus for Henstock-Kurzweil integral, Bulletin of Marathwada Mathematical Society, Vol 14, No 1, June 2013, pp 71-80, ISSN 0976-6049. Anil Pedgaonkar Ph.d theis BAMU,aurangabad india 4310004. Bartle R.G, A Modern theory of Integration, American Mathematical Society, GSM, 32, AMS – 2000 Muldowney Patrick, Modern Theory of random variation, John Wiley & Sons Inc,ISBN-13: 9781118166406 John Wiley & Sons Inc Riecan B, On kurzweil integral in compact topological spaces, Rad Mat 2 ( 1986), pp 151 -163

49. extensions There are no improper integrals. Hk integral is cvclosed for cauchy harnack extensions. All improper riemann integrable functions are Hk integrable. Howeverone can consider total hk integral which extends fundamnetal theorem of calculus even further when F is not continuous. We can have 1/x 2 integrable over R.. And dirac delta function with total integral as 1.The approach may yield different treatment of generalized functions