The purpose of this thesis is to restate the definition of the integral as given by O. Perron, to establish some of the fundamental properties of the Perron integral, and to prove the equivalence between the Perron and Lebesgue integrals in the bounded case.

This dissertation concerns two topics in analysis. The rst section is an exposition of the Henstock-Kurzweil integral leading to a necessary and su cient condition for the change of variables formula to hold, with implications for the change of variables formula for the Lebesgue integral. As a corollary, a necessary and suf- cient condition for the Fundamental Theorem of Calculus to hold for the HK integral is obtained. The second section concerns a challenge raised in a paper by O. Lazarev and E. H. Lieb, where they proved that, given f1….,fn ∈ L1 ([0,1] ; C), there exists a smooth function φ that takes values on the unit circle and annihilates span {f1...., fn}. We give an alternative proof of that fact that also shows the W1,1 norm of φ can be bounded by 5πn + 1. Answering a question raised by Lazarev and Lieb, we show that if p > 1 then there is no bound for the W1,p norm of any such multiplier in terms of the norms of f1...., fn. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection

Thesis (M.S. in Applied Mathematics)--Naval Postgraduate School, September 2002. / Thesis advisor(s): Chris Frenzen, Bard Mansager. Includes bibliographical references (p. 49). Also available online.

In calculus the concept of an infinite derivative – i.e. DF(x) = ±∞ – is seldom studied due to a plethora of complications that arise from this definition. For instance, in this extended sense, algebraic expressions involving derivatives are generally undefined; and two continuous functions possessing identical derivatives at every point of an interval generally differ by a non-constant function. These problems are fundamentally irremediable insofar as calculus is concerned and must therefore be addressed in a more general setting. This is quite difficult since the literature on infinite derivatives is rather sparse and seldom accessible to non-specialists. Therefore we supply a self-contained thesis on continuous functions with infinite derivatives aimed at graduate students with a background in real analysis and measure theory.  Predominately we study continuous primitives which satisfy the Luzin condition (N) by establishing a deep connection with the strong Luzin condition – a weak form of absolute continuity which has its origins in the Henstock–Kurzweil theory of integration. The main result states that a function satisfies the strong Luzin condition if and only if it can be expressed as a sum of two such primitives. Furthermore, we establish some pathological properties of continuous primitives which fail to satisfy the Luzin condition (N).

The aim of this thesis is to build the foundations of generalized ordinary differ- ential equation theory in metric spaces. While differential equations in metric spaces have been studied before, the chosen approach cannot be extended to in- clude more general types of integral equations. We introduce a definition which combines the added generality of metric spaces with the strength of Kurzweil's generalized ordinary differential equations. Additionally, we present existence and uniqueness theorems which offer new results even in the context of Euclidean spaces.

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Since the time of Cauchy, integration theory has in the main been an attempt to regain the Eden of Newton. In that idyllic time [. . . ] derivatives and integrals were [. . . ] different aspects of the same thing. -Peter Bullen, as quoted in [24] The theory of integration has gone through many changes in the past centuries and, in particular, there has been a tension between the Riemann and the Lebesgue approach to integration. Riemann's definition is often the first integral to be introduced in undergraduate studies, while Lebesgue's integral is more powerful but also more complicated and its methods are often postponed until graduate or advanced undergraduate studies. The integral presented in this paper is due to the work of Ralph Henstock and Jaroslav Kurzweil. By a simple exchange of the criterion for integrability in Riemann's definition a powerful integral with many properties of the Lebesgue integral was found. Further, the generalized Riemann integral expands the class of integrable functions with respect to Lebesgue integrals, while there is a characterization of the Lebesgue integral in terms of absolute integrability. As this definition expands the class of functions beyond absolutely integrable functions, some theorems become more cumbersome to prove in contrast to elegant results in Lebesgue's theory and some important properties in composition are lost. Further, it is not as easily abstracted as the Lebesgue integral. Therefore, the generalized Riemann integral should be thought of as a complement to Lebesgue's definition and not as a replacement. / Ända sedan Cauchys tid har integrationsteori i huvudsak varit ett försök att åter finna Newtons Eden. Under den idylliska perioden [. . . ] var derivator och integraler [. . . ] olika sidor av samma mynt.-Peter Bullen, citerad i [24] Under de senaste århundradena har integrationsteori genomgått många förändringar och framförallt har det funnits en spänning mellan Riemanns och Lebesgues respektive angreppssätt till integration. Riemanns definition är ofta den första integral som möter en student pa grundutbildningen, medan Lebesgues integral är kraftfullare. Eftersom Lebesgues definition är mer komplicerad introduceras den först i forskarutbildnings- eller avancerade grundutbildningskurser. Integralen som framställs i det här examensarbetet utvecklades av Ralph Henstock och Jaroslav Kurzweil. Genom att på ett enkelt sätt ändra kriteriet for integrerbarhet i Riemanns definition finner vi en kraftfull integral med många av Lebesgueintegralens egenskaper. Vidare utvidgar den generaliserade Riemannintegralen klassen av integrerbara funktioner i jämförelse med Lebesgueintegralen, medan vi samtidigt erhåller en karaktärisering av Lebesgueintegralen i termer av absolutintegrerbarhet. Eftersom klassen av generaliserat Riemannintegrerbara funktioner är större än de absolutintegrerbara funktionerna blir vissa satser mer omständiga att bevisa i jämforelse med eleganta resultat i Lebesgues teori. Därtill förloras vissa viktiga egenskaper vid sammansättning av funktioner och även möjligheten till abstraktion försvåras. Integralen ska alltså ses som ett komplement till Lebesgues definition och inte en ersättning.

Title: Nonabsolutely convergent integrals Author: Krist'yna Kuncov'a Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jan Mal'y, DrSc., Department of Mathematical Analysis Abstract: In this thesis we develop the theory of nonabsolutely convergent Hen- stock-Kurzweil type packing integrals in different spaces. In the framework of metric spaces we define the packing integral and the uniformly controlled inte- gral of a function with respect to metric distributions. Applying the theory to the notion of currents we then prove a generalization of the Stokes theorem. In Rn we introduce the packing R and R∗ integrals, which are defined as charges - additive functionals on sets of bounded variation. We provide comparison with miscellaneous types of integrals such as R and R∗ integral in Rn or MCα integral in R. On the real line we then study a scale of integrals based on the so called p-oscillation. We show that our indefinite integrals are a.e. approximately differ- entiable and we give comparison with other nonabsolutely convergent integrals. Keywords: Nonabsolutely convergent integrals, BV sets, Henstock-Kurzweil in- tegral, Divergence theorem, Analysis in metric measure spaces 1

Since the introduction of the Riemann integral in the middle of the nineteenth century, integration theory has been subject to significant breakthroughs on a relatively frequent basis. We have now reached a point where integration theory has been thoroughly researched to a point where one has to delve quite deep into a particular subject in order to encounter open conjectures. In education the Riemann integral has for quite some time been the standard integral in elementary analysis courses and as the complexity of these courses incrementally increase the more general Lebesgue integral eventually becomes the standard integral.  Unfortunately, in the transition from the Riemann integral to the Lebesgue integral there are certain topics of pure theoretical interest which to a certain extent are neglected. This is particularly the case for topics regarding the inverse relationship between differential and integral calculus and the integration of exceedingly complicated functions which for example might be of a highly oscillatory nature. From an applied mathematician's point of view, the partial neglection of these topics in the case of highly problematic functions might be justified in the sense that this theory is unnecessary for modeling most problems that appear in nature. From a theoretician's point of view however this negligence is unacceptable. Consequently, there are alternative integrals which give rise to theories which one can use in an attempt to study these aforementioned topics. An example of such an integral is the Henstock–Kurzweil integral, which can be developed in a rather similar manner to that of the Riemann integral.  In this thesis we will develop the Henstock–Kurzweil integral in order to answer some of the questions which to a certain extent are beyond the scope of the Lebesgue integral while using rather basic proof techniques from complex analysis and measure theory. In addition to that we extended various properties of the Lebesgue integral to the Henstock–Kurzweil integral, in particular when it comes to Lebesgue's fundamental theorem of calculus and the basic convergence theorems of the Lebesgue integral.