theories of integration the integrals of riemann lebesgue henstock kurzweil and mcshane

[...]... Fundamental Theorem of Calculus 5.4 Convergence theorems 5.5 The McShane integral as a set function 5.6 The space of McShane integrable functions 5.7 McShane, Henstock- Kurzweil and Lebesgue integrals 5.8 McShane integrals on R" 5.9 Fubini and Tonelli Theorems 5.10 McShane, Henstock- Kurzweil and Lebesgue integrals. .. side of 0 2 2 ~ 1 As in the argument above, it is enough to show that the area of the region bounded by the arc from E t o F and the segments D E and D F is greater than twice the area of the two regions bounded by the arc from E to F and the segments E N , H J and F J More simply, let S' be the region bounded by the arc from E to G and the segments E H and GH and S be the region bounded by the arc... a and xn = b The French mathematician Augustin-Louis Cauchy (1789-1857) studied the area of the region R for continuous functions He approximated the area of the region R by the Cauchy sum Cauchy used the value of the function at the left hand endpoint of each subinterval [xi-l, xi] to generate rectangles with area f (xi-1)(xi- xi-1) The sum of the areas of the rectangles approximate the area of the. .. that is, the area of the region R = ((2, y) : a 5 x 5 b, 0 5 y 5 f (x)} t Figure 1.4 Analogous to the calculation of the area of the circle, we consider approximating the area of the region R by the sums of the areas of rectangles We divide the interval [a,b] into subintervals and use these subintervals for the bases of the rectangles A partition of an interval [a,b] is a finite, ordered set of points... sits on one of its sides Consider the lower right hand corner in the picture below Figure 1.3 Let D be the lower right hand vertex of O4 and let E and F be the points to the left of and above D , respectively, where 0 4 and C meet Let G be midpoint of the arc on C from E to F , and let H and J be the points where the tangent to C at G meets the segments D E and D F , respectively Note that the segment... conditions on the subintervals or sampling points, leads to other, more general integration theories In the Lebesgue theory of integration, the range of the function f is partitioned instead of the domain A representative value, y, is chosen for each subinterval The idea is then to multiply this value by the length of the set of points for which f is approximately equal to y The problem is that this set of points... integral 2.1 Riemann s definition The Riemann integral, defined in 1854 (see [Ril],[Ri2]),was the first of the modern theories of integration and enjoys many of the desirable properties of an integration theory While the most popular integral discussed in introductory analysis texts, the Riemann integral does have serious shortcomings which motivated mathematicians to seek more general integration theories. .. regions in the plane in Chapter 3 The basic idea employed by the ancient Greeks leads in a very natural way to the modern theories of integration, using rectangles instead of triangles to compute the approximating areas For example, let f be a positive 6 Theories of Integration function defined on an interval [a,b] Consider the problem of computing the area of the region under the graph of the function... the Riemann sums as described above, but uses a different condition to control the size of the partition than that employed by Riemann It will be seen that this leads to a very powerful theory more general than the Riemann (or Lebesgue) theory The McShane integral, discussed in Chapter 5 , likewise uses Riemanntype sums The construction of the McShane integral is exactly the same as the Henstock- Kurzweil. .. by using the method of exhaustion Specifically, Archimedes claimed that the area of a circle of radius r is equal to the area of the right triangle with one leg equal to the radius of the circle and the other leg equal to the circumference of the circle We will illustrate the method using modern not a t ion Let C be a circle with radius r and area A Let n be a positive integer, and let In and On be . integral and the Fundamental Theorem of Calculus to motivate the Henstock- Kurzweil integral. We begin the discussion of the Lebesgue integral by establishing the standard convergence theorem for the. discuss versions of the Fundamental The- orem of Calculus for both the Riemann and Lebesgue integrals and give examples showing that the most general form of the Fundamental Theorem of Calculus. integrals in the Henstock- Kurzweil theory. After comparing the Henstock- Kurzweil integral with the Lebesgue integral, we conclude the chapter with a discus- sion of the space of Henstock- Kurzweil