Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

More documents

Recommendations

Info

iv KURZWEIL-HENSTOCK INTEGRAL IN RIESZ SPACES Antonio Boccuto, Beloslav Riean, Marta Vrábelová Abstract: This monograph is concerned with both the theory of the Kurzweil- Henstock integral and the basic facts on Riesz spaces. Moreover even the so-called ipo integral, which has several applications in economy,is illustrated. The aim of this book is two-fold. First, it can be understood as an introductory textbook to the Kurzweil-Henstock integral as well as to some algebraic structures which are important from the viewpoint of applications to integration and probability theory. Second, it discusses some possibilities of further developments including recent results and open problems. CONTRIBUTORS Antonio Boccuto, Doc., Ph.D. Assistant Professor, Dipartimento di Matematica e Informatica, via Vanvitelli, 1 I-06123 Perugia, Italy Email: boccuto@yahoo.it Beloslav Riean, Prof. RNDr., DrSc. Professor, Matej Bel University, Tajovského 40, SK-97401 Banská Bystrica, Slovakia, and Mathematical Institute, Slovak Academy of Science, tefánikova 49, SK- 81473 Bratislava, Slovakia Email: riecan@fpv.umb.sk Marta Vrábelová, Doc. RNDr., CSc. Professor, Constantine the Philosopher University, Tr. A. Hlinku 1, SK-94974 Nitra, Slovakia Email: mvrabelova@ukf.sk Address correspondence to Antonio Boccuto, Dipartimento di Matematica e Informatica, via Vanvitelli, 1 I- 06123 Perugia, Italy, fax +39 075 5855024 Email: boccuto@yahoo.it
Kurzweil-Henstock Integral in Riesz Spaces, 1-24 1 1 Elementary Introduction to Kurzweil-Henstock Integral Abstract In this chapter we introduce the theory of the Kurzweil-Henstock integral for real-valued functions, defined on a bounded interval of the real line. The main properties are illustrated, the Fundamental Theorem of Calculus and some convergence theorems are proved; moreover some examples and exercises are given. 1.1 Introduction In this section we introduce the Kurzweil-Henstock integral (or the gauge integral or the generalized Riemann integral) for real-valued functions f defined on a bounded interval[ ab , ] ⊂ R , only in Subsection 1.9 the interval [ ab , ] is taken unbounded. The integral theory, which we are speaking about, was, independently, introduced by Ralph Henstock (1955) and Jaroslav Kurzweil (1957). This integral is simpler than the Lebesgue integral and as strong as the Denjoy-Perron integral. The definition is similar to the definition of the Riemann integral. This chapter does not fetch new results. Its aim is that the reader became familiar with the Kurzweil-Henstock integral theory. Our intention is especially to point out some differences between the Kurzweil-Henstock integral and the Riemann integral. The proofs and the solutions are given in details and the section is readable for a reader with an elementary knowledge of the calculus. For a deeper study of this theory there are a few interesting books. We mention, for example, the books written by Robert McLeod [188], Lee Peng Yee and Rudolf Výborný [170], and Charles Swartz [254]. A division of a compact interval [ ab , ] is a finite, ordered sequence of points a= a0 < a1 < …< an= b. By choosing a number ti∈ [ ai−1, ai] for i = 12 , , …, n a partition P of [ ab , ] is defined, P = {([ ai−1, ai] , ti) : i = 1, 2 , …, n} . If δ > 0, then a partition P with ti − δ < ai−1≤ti ≤ ai < ti + δ for i = 12 , , …, n is called a δ -fine partition of [ ab , ] . Lets suppose that f is a bounded real-valued function defined on[ ab , ] . A Riemann sum for the P = {([ a , a ] , t ) : i = 1, 2 , …, n} of the interval [ ab , ] is the number function f and the partition i−1 n i i ∑ ∑ f = f( ti)( ai − ai−1) . P i= 1 Generally, a partition P of [ ab , ] is a set of the type P = {( Ei, ti) : i = 1, 2 , …, n} , where i E are non-overlapping, closed intervals, n ti∈ Ei for i = 12 , , …, n and U i= 1 Ei = [ a, b] . The Riemann sum is the quantity n ∑ ∑ f = f( t ) |E | , P i= 1 i i where |E i | is the length of the interval E i , i = 1, …, n. Antonio Boccuto / Beloslav Riečan / Marta Vrábelová All rights reserved - © 2009 Bentham Science Publishers Ltd.
Page 2 and 3: Kurzweil-Henstock Integral in Riesz
Page 4 and 5: 7 Improper Integral 84 7.1 Real val
Page 6 and 7: ii PREFACE Kurzweil and Henstock’
Page 10 and 11: 2 Kurzweil-Henstock Integral in Rie
Page 14 and 15: Elementary Theory of Riesz Spaces K
Page 16 and 17: Kurzweil-Henstock Integral with Val
Page 18 and 19: 52 Kurzweil-Henstock Integral in Ri
Page 20 and 21: 54 Kurzweil-Henstock Integral in Ri
Page 22 and 23: Kurzweil-Henstock Integral in Topol
Page 24 and 25: 6 Convergence Theorems Kurzweil-Hen
Page 26 and 27: Convergence Theorems Kurzweil-Henst
Page 28 and 29: Improper Integral Kurzweil-Henstock
Page 30 and 31: 8 Choquet and ipo Integrals Kurzwei
Page 32 and 33: Choquet and ipo Integrals Kurzweil-
Page 34 and 35: (SL)-Integral Kurzweil-Henstock Int
Page 36 and 37: 166 Kurzweil-Henstock Integral in R
Page 38 and 39: 168 Kurzweil-Henstock Integral in R
Page 40 and 41: Applications in Multivalued Logic K
Page 44 and 45: Applications in Probability Theory
Page 46 and 47: Integration in Metric Semigroups Ku
Page 50 and 51: References Kurzweil-Henstock Integr
Page 52 and 53: Index Kurzweil-Henstock Integral in

Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?