DOCSLIB.ORG

Lecture 1: Introduction to Integration Theory and Bounded Variation What

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 1: Introduction to Integration Theory and Bounded Variation What Lecture 1: Introduction to integration theory and bounded variation What is this course about? Integration theory. The first question you might have is why there is anything you need to learn about integration. You took a class called Math 1a and learned something like this: Given a bounded real-valued function f on an interval [a; b], we want to define Z b f(x)dx: a So what we do is that we partition the interval [a; b]. A partition P of [a; b] is a finite set P = fx0; x1; : : : ; xng with a = x0 < x1 < : : : < xn = b: For every partition P , we define upper and lower Riemann sums: n X UP (f) = (xj − xj−1) sup f(x); j=1 x2[xj−1;xj ] and n X LP (f) = (xj − xj−1) inf f(x); x2[x ;x ] j=1 j−1 j We that a partition Q refines a partition P if P ⊂ Q and if Q refines P we have the inequalities UQ(f) ≤ UP (f); and LP (f) ≤ LQ(f): (Why?) Then we define upper and lower integrals by b Ua(f) = inf UP (f); P and b La(f) = sup LP (f): P 1 b b If the two numbers, Ua(f) and La(f) match, then we say that f is integrable and denote this number by Z b f(x)dx: a Otherwise, we say that f is not integrable. The integral above is called the Riemann integral. Are there any drawbacks to this set-up. One thing we could worry about is that we haven't entirely worked out which functions are Riemann integrable and which are not. [We could do that this quarter]. But we know something about it. For instance: Proposition 1 A function f which is continuous on [a; b] is Riemann integrable. Proof: Since [a; b] is compact, we know that f is uniformly continuous on [a; b]. Hence for every > 0, there is δ > 0 so that whenever x; y 2 [a; b] and jx − yj < δ, it must be that jf(x) − f(y)j < . Then if P is a partition so that jxj − xj−1j < δ for every j, we must have UP (f) − LP (f) < (b − a): (Why?) Thus for every > 0, we have b b Ua(f) − La(f) < (b − a): We conclude the difference must be zero, hence f is Riemann integrable. But continuity is not really a necessary condition. For instance, a bounded function f with just finitely many discontinuities is still Riemann integrable. (Why?) A slightly worse f is given by 1 if x 2 Q; f(x) = 0 if x2 = Q. This is a disaster. Since the rationals, Q, are dense in the reals, every point is a point of discontinuity of f, and in any subinterval of [a; b], we find a point where f is 1 and a point where f is 0. We get b Ua(f) = b − a; while b La(f) = 0: 2 Thus the function f is not integrable. What's really wrong with that f? The rationals Q are really a fairly small set. They're countable whereas the reals are not. Everywhere but on the rationals, the function f is zero. Why should the rationals have so much weight in determining the integral. (Answer: Because it's a Riemann integral, location, location, location.) A more practical minded person might view such a question as nonsense. A function like f can't really appear in nature. How can you even measure whether a number is rational or irrational? These practical people might think that we should be interested only in much \nicer" functions and what identities we can discover about their integrals. For instance, if we take f(x) instead to be a continuous function on [a; b] and let F (x) be its antiderivative then we have Z b f(x)dx = F (b) − F (a); a the fundamental theorem of calculus. (Secretly, this is some identity about telescoping sums.) We can get still fancier identities out of this. To wit: Proposition 2 (Integration by parts) Let f and g be once continuously differen- tiable functions on [a; b]. Then Z b Z b f(x)g0(x)dx = − f 0(x)g(x)dx + f(b)g(b) − f(a)g(a): a a Proof We use the fundamental theorem of calculus and the product rule for derivatives to get Z b [f(x)g0(x) + f 0(x)g(x)]dx = f(b)g(b) − f(a)g(a): a Integration by parts is definitely cool. It is a somewhat deeper statement about telescoping sums than the fundamental theorem and lets us derive lots of identities and estimates. Is continuity of the derivative really necessary for integration by parts to work? This is in fact what we'll be focusing on for the first two weeks of the course. But why do we care? Is it of any practical importance to know which functions integration by parts works for? What this course is largely about is the question of what estimates (inequalities) depend on. When we prove the integration by parts identity, we are actually proving inequalities because both sides of the equation are limits. When we have a theorem, we know the limits are equal, but we might care how fast they're converging to 3 one another. What does that depend on? Continuity is a complicated thing to quantify. Even accepting that the continuity of derivatives has to be uniform on [a; b], the input into any one of our proofs is really how δ depends on . We might like everything to be based on a simple quantity instead. Part of what we're going to see over the course of this term (and studying analysis in general) is that there's perfect universal definition of integral that is best for every situation. Rather, we should introduce different integrals for different purposes and usually there's a nice class of functions each one works for. We should try to see how everything depends on the definition of the class. But the simpler and more general the class of functions is, the better. We begin the course by introducing a class of functions for which we'll see that integration by parts works very well. These are the functions of bounded variation. We proceed to define them: Given f :[a; b] −! R; and P = fx); x1; : : : ; xng a partition of [a; b] as before, we define the variation of f with respect to the partition P by N X V (f; P ) = jf(xj) − f(xj−1)j: j=1 It is easy to see that if Q refines P then V (f; Q) ≥ V (f; P ): We define the total variation of f on [a; b] by b Va = sup V (f; P ): P b If Va (f) < 1, we say that f is of bounded variation on [a; b], or sometimes write f 2 BV ([a; b]): To work in the class of functions of bounded variation really means to use these definitions, that is to make all estimates depend only on total variations. I'll close this lecture with some examples. Example 1: Let f be a monotone increasing function on [a; b]. That is, suppose that if x < y, we have f(x) ≤ f(y). Then the sum defining V (f; P ) telescopes so that we have V (f; P ) = f(b) − f(a); 4 b regardless of P . Then this quantity also gives the total variation Va f. Example 2: You're probably dying to know what happens with \nice" functions. Let f be once continuously differentiable on [a; b]. Then Z b b 0 Va f = jf (x)jdx: a How do we know. Pick a partition P . Then by the mean value theorem for each part [xj−1; xj], there is a point cj in the interval so that 0 f (cj)(xj − xj−1) = f(xj) − f(xj−1): Thus the variation V (f; P ) is a Riemann sum of jf 0j on the partition P . As we take the sup of variations we arrive at the integral. Thus using the theory of bounded variation on just nice functions means tying one hand behind our backs and making everything just depend on total variations, that is on integrals of absolute values of derivatives. Finally: Example 3 Let's recall our old enemy 1 if x 2 Q; f(x) = 0 if x2 = Q. We can pick a partition P where the xj's alternate between being rational and irrational. This makes the variation arbitrarily large. We see that this function is not of bounded variation, which is just the tip of the iceberg of what is wrong with it. 5.
Load more

Recommended publications
  • Vector Measures with Variation in a Banach Function Space
    VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE O. BLASCO Departamento de An´alisis Matem´atico, Universidad de Valencia, Doctor Moliner, 50 E-46100 Burjassot (Valencia), Spain E-mail:[email protected] PABLO GREGORI Departament de Matem`atiques, Universitat Jaume I de Castell´o, Campus Riu Sec E-12071 Castell´o de la Plana (Castell´o), Spain E-mail:[email protected] Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the K¨othe-Bochner function space defined as the set of measurable functions f :Ω→ X such that the nonnegative functions fX :Ω→ [0, ∞) are in the lattice E. The notion of E-variation of a measure —which allows to recover the p- variation (for E = Lp), Φ-variation (for E = LΦ) and the general notion introduced by Gresky and Uhl— is introduced. The space of measures of bounded E-variation VE (X) is then studied. It is shown, amongother thingsand with some restriction ∗ ∗ of absolute continuity of the norms, that (E(X)) = VE (X ), that VE (X) can be identified with space of cone absolutely summingoperators from E into X and that E(X)=VE (X) if and only if X has the RNP property. 1. Introduction The concept of variation in the frame of vector measures has been fruitful in several areas of the functional analysis, such as the description of the duality of vector-valued function spaces such as certain K¨othe-Bochner function spaces (Gretsky and Uhl10, Dinculeanu7), the reformulation of operator ideals such as the cone absolutely summing operators (Blasco4), and the Hardy spaces of harmonic function (Blasco2,3).
    [Show full text]
  • The Total Variation Flow∗
    THE TOTAL VARIATION FLOW∗ Fuensanta Andreu and Jos´eM. Maz´on† Abstract We summarize in this lectures some of our results about the Min- imizing Total Variation Flow, which have been mainly motivated by problems arising in Image Processing. First, we recall the role played by the Total Variation in Image Processing, in particular the varia- tional formulation of the restoration problem. Next we outline some of the tools we need: functions of bounded variation (Section 2), par- ing between measures and bounded functions (Section 3) and gradient flows in Hilbert spaces (Section 4). Section 5 is devoted to the Neu- mann problem for the Total variation Flow. Finally, in Section 6 we study the Cauchy problem for the Total Variation Flow. 1 The Total Variation Flow in Image Pro- cessing We suppose that our image (or data) ud is a function defined on a bounded and piecewise smooth open set D of IRN - typically a rectangle in IR2. Gen- erally, the degradation of the image occurs during image acquisition and can be modeled by a linear and translation invariant blur and additive noise. The equation relating u, the real image, to ud can be written as ud = Ku + n, (1) where K is a convolution operator with impulse response k, i.e., Ku = k ∗ u, and n is an additive white noise of standard deviation σ. In practice, the noise can be considered as Gaussian. ∗Notes from the course given in the Summer School “Calculus of Variations”, Roma, July 4-8, 2005 †Departamento de An´alisis Matem´atico, Universitat de Valencia, 46100 Burjassot (Va- lencia), Spain, [email protected], [email protected] 1 The problem of recovering u from ud is ill-posed.
    [Show full text]
  • Total Variation Deconvolution Using Split Bregman
    Published in Image Processing On Line on 2012{07{30. Submitted on 2012{00{00, accepted on 2012{00{00. ISSN 2105{1232 c 2012 IPOL & the authors CC{BY{NC{SA This article is available online with supplementary materials, software, datasets and online demo at http://dx.doi.org/10.5201/ipol.2012.g-tvdc 2014/07/01 v0.5 IPOL article class Total Variation Deconvolution using Split Bregman Pascal Getreuer Yale University ([email protected]) Abstract Deblurring is the inverse problem of restoring an image that has been blurred and possibly corrupted with noise. Deconvolution refers to the case where the blur to be removed is linear and shift-invariant so it may be expressed as a convolution of the image with a point spread function. Convolution corresponds in the Fourier domain to multiplication, and deconvolution is essentially Fourier division. The challenge is that since the multipliers are often small for high frequencies, direct division is unstable and plagued by noise present in the input image. Effective deconvolution requires a balance between frequency recovery and noise suppression. Total variation (TV) regularization is a successful technique for achieving this balance in de- blurring problems. It was introduced to image denoising by Rudin, Osher, and Fatemi [4] and then applied to deconvolution by Rudin and Osher [5]. In this article, we discuss TV-regularized deconvolution with Gaussian noise and its efficient solution using the split Bregman algorithm of Goldstein and Osher [16]. We show a straightforward extension for Laplace or Poisson noise and develop empirical estimates for the optimal value of the regularization parameter λ.
    [Show full text]
  • Guaranteed Deterministic Bounds on the Total Variation Distance Between Univariate Mixtures
    Guaranteed Deterministic Bounds on the Total Variation Distance between Univariate Mixtures Frank Nielsen Ke Sun Sony Computer Science Laboratories, Inc. Data61 Japan Australia [email protected] [email protected] Abstract The total variation distance is a core statistical distance between probability measures that satisfies the metric axioms, with value always falling in [0; 1]. This distance plays a fundamental role in machine learning and signal processing: It is a member of the broader class of f-divergences, and it is related to the probability of error in Bayesian hypothesis testing. Since the total variation distance does not admit closed-form expressions for statistical mixtures (like Gaussian mixture models), one often has to rely in practice on costly numerical integrations or on fast Monte Carlo approximations that however do not guarantee deterministic lower and upper bounds. In this work, we consider two methods for bounding the total variation of univariate mixture models: The first method is based on the information monotonicity property of the total variation to design guaranteed nested deterministic lower bounds. The second method relies on computing the geometric lower and upper envelopes of weighted mixture components to derive deterministic bounds based on density ratio. We demonstrate the tightness of our bounds in a series of experiments on Gaussian, Gamma and Rayleigh mixture models. 1 Introduction 1.1 Total variation and f-divergences Let ( R; ) be a measurable space on the sample space equipped with the Borel σ-algebra [1], and P and QXbe ⊂ twoF probability measures with respective densitiesXp and q with respect to the Lebesgue measure µ.
    [Show full text]
  • Calculus of Variations
    MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 16.323 Lecture 5 Calculus of Variations • Calculus of Variations • Most books cover this material well, but Kirk Chapter 4 does a particularly nice job. • See here for online reference. x(t) x*+ αδx(1) x*- αδx(1) x* αδx(1) −αδx(1) t t0 tf Figure by MIT OpenCourseWare. Spr 2008 16.323 5–1 Calculus of Variations • Goal: Develop alternative approach to solve general optimization problems for continuous systems – variational calculus – Formal approach will provide new insights for constrained solutions, and a more direct path to the solution for other problems. • Main issue – General control problem, the cost is a function of functions x(t) and u(t). � tf min J = h(x(tf )) + g(x(t), u(t), t)) dt t0 subject to x˙ = f(x, u, t) x(t0), t0 given m(x(tf ), tf ) = 0 – Call J(x(t), u(t)) a functional. • Need to investigate how to find the optimal values of a functional. – For a function, we found the gradient, and set it to zero to find the stationary points, and then investigated the higher order derivatives to determine if it is a maximum or minimum. – Will investigate something similar for functionals. June 18, 2008 Spr 2008 16.323 5–2 • Maximum and Minimum of a Function – A function f(x) has a local minimum at x� if f(x) ≥ f(x �) for all admissible x in �x − x�� ≤ � – Minimum can occur at (i) stationary point, (ii) at a boundary, or (iii) a point of discontinuous derivative.
    [Show full text]
  • Calculus of Variations in Image Processing
    Calculus of variations in image processing Jean-François Aujol CMLA, ENS Cachan, CNRS, UniverSud, 61 Avenue du Président Wilson, F-94230 Cachan, FRANCE Email : [email protected] http://www.cmla.ens-cachan.fr/membres/aujol.html 18 september 2008 Note: This document is a working and uncomplete version, subject to errors and changes. Readers are invited to point out mistakes by email to the author. This document is intended as course notes for second year master students. The author encourage the reader to check the references given in this manuscript. In particular, it is recommended to look at [9] which is closely related to the subject of this course. Contents 1. Inverse problems in image processing 4 1.1 Introduction.................................... 4 1.2 Examples ....................................... 4 1.3 Ill-posedproblems............................... .... 6 1.4 Anillustrativeexample. ..... 8 1.5 Modelizationandestimator . ..... 10 1.5.1 Maximum Likelihood estimator . 10 1.5.2 MAPestimator ................................ 11 1.6 Energymethodandregularization. ....... 11 2. Mathematical tools and modelization 14 2.1 MinimizinginaBanachspace . 14 2.2 Banachspaces.................................... 14 2.2.1 Preliminaries ................................. 14 2.2.2 Topologies in Banach spaces . 17 2.2.3 Convexity and lower semicontinuity . ...... 19 2.2.4 Convexanalysis................................ 23 2.3 Subgradient of a convex function . ...... 23 2.4 Legendre-Fencheltransform: . ....... 24 2.5 The space of funtions with bounded variation . ......... 26 2.5.1 Introduction.................................. 26 2.5.2 Definition ................................... 27 2.5.3 Properties................................... 28 2.5.4 Decomposability of BV (Ω): ......................... 28 2.5.5 SBV ...................................... 30 2.5.6 Setsoffiniteperimeter . 32 2.5.7 Coarea formula and applications .
    [Show full text]
  • Chapter 2. Functions of Bounded Variation §1. Monotone Functions
    Chapter 2. Functions of Bounded Variation §1. Monotone Functions Let I be an interval in IR. A function f : I → IR is said to be increasing (strictly increasing) if f(x) ≤ f(y)(f(x) < f(y)) whenever x, y ∈ I and x < y. A function f : I → IR is said to be decreasing (strictly decreasing) if f(x) ≥ f(y)(f(x) > f(y)) whenever x, y ∈ I and x < y. A function f : I → IR is said to be monotone if f is either increasing or decreasing. In this section we will show that a monotone function is differentiable almost every- where. For this purpose, we first establish Vitali covering lemma. Let E be a subset of IR, and let Γ be a collection of closed intervals in IR. We say that Γ covers E in the sense of Vitali if for each δ > 0 and each x ∈ E, there exists an interval I ∈ Γ such that x ∈ I and `(I) < δ. Theorem 1.1. Let E be a subset of IR with λ∗(E) < ∞ and Γ a collection of closed intervals that cover E in the sense of Vitali. Then, for given ε > 0, there exists a finite disjoint collection {I1,...,IN } of intervals in Γ such that ∗ N λ E \ ∪n=1In < ε. Proof. Let G be an open set containing E such that λ(G) < ∞. Since Γ is a Vitali covering of E, we may assume that each I ∈ Γ is contained in G. We choose a sequence (In)n=1,2,... of disjoint intervals from Γ recursively as follows.
    [Show full text]
  • Section 3.5 of Folland’S Text, Which Covers Functions of Bounded Variation on the Real Line and Related Topics
    REAL ANALYSIS LECTURE NOTES: 3.5 FUNCTIONS OF BOUNDED VARIATION CHRISTOPHER HEIL 3.5.1 Definition and Basic Properties of Functions of Bounded Variation We will expand on the first part of Section 3.5 of Folland's text, which covers functions of bounded variation on the real line and related topics. We begin with functions defined on finite closed intervals in R (note that Folland's ap- proach and notation is slightly different, as he begins with functions defined on R and uses TF (x) instead of our V [f; a; b]). Definition 1. Let f : [a; b] ! C be given. Given any finite partition Γ = fa = x0 < · · · < xn = bg of [a; b], set n SΓ = jf(xi) − f(xi−1)j: i=1 X The variation of f over [a; b] is V [f; a; b] = sup SΓ : Γ is a partition of [a; b] : The function f has bounded variation on [a; b] if V [f; a; b] < 1. We set BV[a; b] = f : [a; b] ! C : f has bounded variation on [a; b] : Note that in this definition we are considering f to be defined at all poin ts, and not just to be an equivalence class of functions that are equal a.e. The idea of the variation of f is that is represents the total vertical distance traveled by a particle that moves along the graph of f from (a; f(a)) to (b; f(b)). Exercise 2. (a) Show that if f : [a; b] ! C, then V [f; a; b] ≥ jf(b) − f(a)j.
    [Show full text]
  • Measure Algebras and Functions of Bounded Variation on Idempotent Semigroupso
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 163, January 1972 MEASURE ALGEBRAS AND FUNCTIONS OF BOUNDED VARIATION ON IDEMPOTENT SEMIGROUPSO BY STEPHEN E. NEWMAN Abstract. Our main result establishes an isomorphism between all functions on an idempotent semigroup S with identity, under the usual addition and multiplication, and all finitely additive measures on a certain Boolean algebra of subsets of S, under the usual addition and a convolution type multiplication. Notions of a function of bounded variation on 5 and its variation norm are defined in such a way that the above isomorphism, restricted to the functions of bounded variation, is an isometry onto the set of all bounded measures. Our notion of a function of bounded variation is equivalent to the classical notion in case S is the unit interval and the "product" of two numbers in S is their maximum. 1. Introduction. This paper is motivated by the pair of closely related Banach algebras described below. The interval [0, 1] is an idempotent semigroup when endowed with the operation of maximum multiplication (.vj = max (x, y) for all x, y in [0, 1]). We let A denote the Boolean algebra (or set algebra) consisting of all finite unions of left-open, right-closed intervals (including the single point 0) contained in the interval [0, 1]. The space M(A) of all bounded, finitely additive measures on A is a Banach space with the usual total variation norm. The nature of the set algebra A and the idempotent multiplication given the interval [0, 1] make it possible to define a convolution multiplication in M(A) which makes it a Banach algebra.
    [Show full text]
  • On the Convolution of Functions of Λp – Bounded Variation and a Locally
    IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 8, Issue 3 (Sep. - Oct. 2013), PP 70-74 www.iosrjournals.org On the Convolution of Functions of p – Bounded Variation and a Locally Compact Hausdorff Topological Group S. M. Nengem1, D. Samaila1 and M. Solomon1 1 Department of Mathematical Sciences, Adamawa State University, P. M. B. 25, Mubi, Nigeria Abstract: The smoothness-increasing operator “convolution” is well known for inheriting the best properties of each parent function. It is also well known that if f L1 and g is a Bounded Variation (BV) function, then f g inherits the properties from the parent’s spaces. This aspect of BV can be generalized in many ways and many generalizations are obtained. However, in this paper we introduce the notion of p – Bounded variation function. In relation to that we show that the convolution of two functions f g is the inverse Fourier transforms of the two functions. Moreover, we prove that if f, g BV(p)[0, 2], then f g BV(p)[0, 2], and that on any locally compact Abelian group, a version of the convolution theorem holds. Keywords: Convolution, p-Bounded variation, Fourier transform, Abelian group. I. Introduction In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated.
    [Show full text]
  • Fonctions on Bounded Variations in Hilbert Spaces
    Fonctions on bounded variations in Hilbert spaces Giuseppe Da Prato (SNS, Pisa) Newton Institute, March 31, 2010 Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Introduction We recall that a function u : Rn ! R is said to be of bounded variation (BV) if there exists an n-dimensional vector measure Du with finite total variation such that Z Z 1 u(x)divF(x)dx = − hF(x); Du(dx)i; 8 F 2 C0 (H; H): H H The set of all BV functions is denoted by BV (Rn). Moreover, the following result holds, see E. De Giorgi, Ann. Mar. Pura Appl. 1954. Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces Theorem 1 Letu 2 L1(Rn). Then (i) , (ii). (i)u 2 BV (Rn). (ii) We have Z lim jDTt u(x)jdx < 1; (1) t!0 H whereT t is the heat semigroup Z 2 1 − jx−yj Tt u(x) := p e 4t u(y)dy: n 4πt R Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces As well known functions from BV (Rn) arise in many mathematical problems as for instance: finite perimeter sets, surface integrals, variational problems in different models in elasto plasticity, image segmentation, and so on. Recently there is also an increasing interest in studyingBV functions in general Banach or Hilbert spaces in order to extend the concepts above in an infinite dimensional situation. Giuseppe Da Prato (SNS, Pisa) Fonctions on bounded variations in Hilbert spaces A definition of BV function in an abstract Wiener space, using a Gaussian measure µ and the corresponding Dirichlet form, has been given by M.
    [Show full text]
  • FUNCTIONS of BOUNDED VARIATION 1. Introduction in This Paper We Discuss Functions of Bounded Variation and Three Related Topics
    FUNCTIONS OF BOUNDED VARIATION NOELLA GRADY Abstract. In this paper we explore functions of bounded variation. We discuss properties of functions of bounded variation and consider three re- lated topics. The related topics are absolute continuity, arc length, and the Riemann-Stieltjes integral. 1. Introduction In this paper we discuss functions of bounded variation and three related topics. We begin by defining the variation of a function and what it means for a function to be of bounded variation. We then develop some properties of functions of bounded variation. We consider algebraic properties as well as more abstract properties such as realizing that every function of bounded variation can be written as the difference of two increasing functions. After we have discussed some of the properties of functions of bounded variation, we consider three related topics. We begin with absolute continuity. We show that all absolutely continuous functions are of bounded variation, however, not all continuous functions of bounded variation are absolutely continuous. The Cantor Ternary function provides a counter example. The second related topic we consider is arc length. Here we show that a curve has a finite length if and only if it is of bounded variation. The third related topic that we examine is Riemann-Stieltjes integration. We examine the definition of the Riemann-Stieltjes integral and see when functions of bounded variation are Riemann-Stieltjes integrable. 2. Functions of Bounded Variation Before we can define functions of bounded variation, we must lay some ground work. We begin with a discussion of upper bounds and then define partition.
    [Show full text]