DOCSLIB.ORG

MEASURE THEORY LECTURE NOTES XI. Complex Measures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MEASURE THEORY LECTURE NOTES XI. Complex Measures MEASURE THEORY LECTURE NOTES JANUSZ ADAMUS XI. Complex Measures Definition 1. Let M be a σ-algebra on a set X.A complex measure on M is a complex-valued function µ : M! C satisfying [ X (1) µ( En) = µ(En) ; n n for every denumerable collection fEngn of pairwise disjoint elements of M. Remark 2. 1. It follows from the definition that µ(?) = 0. Indeed, we have ? = ? t ? and hence µ(?) = 2µ(?), which in light of µ(?) 2 C implies µ(?) = 0. 2. Since the left side of (1) is a complex number, it follows that the series on the right side of (1) is always convergent. Moreover, the sum of the series is independent of the ordering of the sets En, and hence the series is absulutely convergent. Example 3. The following is a model example of a complex measure. By the Radon-Nikodym Theorem below, all complex measures arise in this way modulo a set of measure zero. Let (X; M; µ) be a σ-finite measure space, and let h : X ! C be an integrable function. (Recall that a complex-valued function f = u + iv, where u and v are real-valued, is called integrable when both u and v are integrable as real-valued functions. In this case, one defines R f := R u + i R v.) Then, the function ν : M! C defined as Z (2) ν(A) := h dµ A is a complex measure. The proof is an elementary exercise. Definition 4. Let µ : M! C be a complex measure on a σ-algebra M. The function jµj : M! R defined as 1 1 X [ jµj(E) := supf jµ(En)j : fEngn ⊂ M;E = En;Ej \ Ek = ? for j 6= kg n=1 n=1 is called the total variation measure of µ. Remark 5. Observe that we always have jµj(A) ≥ jµ(A)j, for every A 2 M. Indeed, the family fA; ?; ?;:::g forms a measurable partition of A, and hence 1 X jµj(A) ≥ jµ(A)j + jµ(?)j = jµ(A)j : n=2 1 2 JANUSZ ADAMUS Exercise 6. For a signed measure µ on a σ-algebra M, one defines the total variation measure of µ as jµj := µ+ + µ−, where µ+ and µ− are the unique positive measures from the Jordan Decomposition Theorem for µ. Note that a real-valued (i.e., finite) signed measure µ may be regarded as a complex measure. Prove that if µ is a real-valued signed measure, then the two definitions of jµj coincide. Theorem 7. The total variation jµj of a complex measure µ on a σ-algebra M is a positive measure on M. 1 Proof. Let fEngn=1 ⊂ M be an arbitrary collection of pairwise disjoint measurable S sets. Set E := n En. We want to show that 1 X jµj(E) = jµj(En) : n=1 To this end, for every n 2 Z+, choose an arbitrary real number tn satisfying jµj(En) > tn. Then, by definition of jµj, there exists for every n a measurable 1 P1 partition fAnkgk=1 of En such that k=1 jµ(Ank)j ≥ tn. Since the (countable) 1 family fAnkgn;k=1 forms a partition of E, we get that 1 1 1 ! 1 X X X X jµj(E) ≥ jµ(Ank)j = jµ(Ank)j ≥ tn : n;k=1 n=1 k=1 n=1 (Note that above we used the absolute convergence of the series in question to rearrange its terms without altering the sum.) 1 Taking supremum over all sequences (tn)n=1 as above, we get 1 X jµj(E) ≥ jµj(En) : n=1 1 For the proof of the opposite inequality, let fAkgk=1 ⊂ M be another arbitrary 1 partition of E. Then, for every k, fAk \ Engn=1 is a measurable partition of Ak, 1 and for every n, fAk \ Engk=1 is a measurable partition of En. Thus, again by absolute convergence, we get 1 1 1 1 1 ! X X X X X jµ(Ak)j = µ(Ak \ En) ≤ jµ(Ak \ En)j k=1 k=1 n=1 k=1 n=1 1 1 ! 1 X X X = jµ(Ak \ En)j ≤ jµj(En) : n=1 k=1 n=1 Taking supremum over all countable measurable partitions fAkgk of E, we get P1 jµj(E) ≤ n=1 jµj(En). This proves countable additivity of jµj. The equality jµj(?) = 0 follows from µ(?) = 0 and the fact that the only countable measurable partition of ? consists of copies of ?. Perhaps even more interestingly, the total variation of any complex measure is a finite positive measure, as the following theorem shows. Theorem 8. If µ is a complex measure on a measurable space (X; M), then the total variation measure jµj satisfies jµj(X) < +1. We shall first establish an auxiliary lemma, which is a somewhat surprising complex analysis result of independent interest. MATH4122B/9022B - INTRODUCTION TO MEASURE THEORY 3 Lemma 9. Given any collection fz1; : : : ; zN g of (not necessarily pairwise distinct) N complex numbers, there exists an index subset S ⊂ f1;:::;Ng such that N X 1 X z ≥ · jz j : n π n n2S n=1 iαn Proof. For 1 ≤ n ≤ N, let αn 2 (−π; π] be such that zn = jznje . For # 2 [−π; π], let S(#) := fn 2 f1;:::;Ng : cos(αn − #) > 0g : Then, 0 1 X −i# X X −i# X i(αn−#) j znj = je j · j znj = j e znj ≥ Re @ jznje A n2S(#) n2S(#) n2S(#) n2S(#) N X X + = jznj cos(αn − #) = jznj cos (αn − #) ; n2S(#) n=1 where, as usual, cos+ = maxfcos; 0g. Now, choose #0 2 [−π; π] so as to maximize the latter sum, and set S := S(#0). By the Mean Value Theorem for Riemann integral, we have N N ! X 1 Z π X jz j cos+(α − # ) ≥ jz j cos+(α − #) d# ; n n 0 2π n n n=1 −π n=1 and hence N ! X 1 Z π X j z j ≥ jz j cos+(α − #) d# n 2π n n n2S −π n=1 N N N X 1 Z π X 1 1 X = jz j · cos+(α − #) d# = (jz j · ) = · jz j ; n 2π n n π π n n=1 −π n=1 n=1 where the penultimate equality follows from the fact that cos+ is periodic with R π + period 2π, and hence −π cos (αn − #) d# = 2 independently of the choice of αn. Proof of Theorem 8. Suppose first that, for some E 2 M, we have jµj(E) = +1. Set t := π(1 + jµ(E)j). (Of course, t < +1, since µ is a complex measure.) Since jµj(E) = +1, then by definition of jµj there exist a measurable partition 1 fEngn=1 of E and a positive integer N such that N X jµ(En)j > t : n=1 Applying Lemma 9 with zn := µ(En), we conclude that there is a measurable set A ⊂ E (namely, the union of some of the E1;:::;EN ) such that t jµ(A)j > ≥ 1 : π Moreover, for B := E n A, we also have t jµ(B)j = jµ(E n A)j = jµ(E) − µ(A)j ≥ jµ(A)j − jµ(E)j > − jµ(E)j = 1 : π 4 JANUSZ ADAMUS We thus have E = A t B, with jµ(A)j > 1 and jµ(B)j > 1. By additivity of jµj, at least one of the A; B must be of infinite jµj-measure. 1 Now, if jµj(X) = +1, we construct recursively an infinite sequence (An)n=1 ⊂ M of pairwise disjoint sets with jµ(An)j > 1 for all n, as follows: By the first part of the proof, we can partition X into A1 and B1 such that jµ(A1)j > 1 and jµj(B1) = +1. Having defined A1;:::;Ak and B1;:::;Bk, partition Bk into Ak+1 and Bk+1 such that jµ(Ak+1)j > 1 and jµj(Bk+1) = +1. S1 P1 Then, by additivity of µ, we have µ( n=1 An) = n=1 µ(An). In particular, the P1 series n=1 µ(An) is absolutely convergent. Hence, by the Basic Divergence Test, lim jµ(An)j = 0, contradicting the choice of the An. n!1 Exercise 10. (a) Show that if µ, ν are complex measures on a σ-algebra M, then so are µ + ν and c · µ for any c 2 C (where (µ + ν)(E) = µ(E) + ν(E) and (cµ)(E) = c · µ(E) for E 2 M). Therefore the set of all complex measures on M forms a complex vector space. (b) Prove that the function kµk := jµj(X) defines a norm on that vector space. Absolute Continuity. Definition 11. Let µ be a positive measure on a σ-algebra M, and let λ be an arbitrary measure on M. We say that λ is absolutely continuous with respect to µ, and write λ µ, when λ(E) = 0 for all E 2 M with µ(E) = 0. For the following proposition, recall that a (positive, signed, or complex) measure µ is said to be concentrated on a measurable set A, when µ(E) = 0 for every measurable E ⊂ Ac. Proposition 12. Suppose λ, λ1; λ2 are arbitrary measures on a σ-algebra M, and µ is a positive measure on M. Then: (i) If λ is concentrated on A 2 M, then so is jλj. (ii) If λ1?λ2, then jλ1j?jλ2j. (iii) If λ1?µ and λ2?µ, then (λ1 + λ2)?µ. (iv) If λ1 µ and λ2 µ, then (λ1 + λ2) µ.
Load more

Recommended publications
  • Complex Measures 1 11
    Tutorial 11: Complex Measures 1 11. Complex Measures In the following, (Ω, F) denotes an arbitrary measurable space. Definition 90 Let (an)n≥1 be a sequence of complex numbers. We a say that ( n)n≥1 has the permutation property if and only if, for ∗ ∗ +∞ 1 all bijections σ : N → N ,theseries k=1 aσ(k) converges in C Exercise 1. Let (an)n≥1 be a sequence of complex numbers. 1. Show that if (an)n≥1 has the permutation property, then the same is true of (Re(an))n≥1 and (Im(an))n≥1. +∞ 2. Suppose an ∈ R for all n ≥ 1. Show that if k=1 ak converges: +∞ +∞ +∞ + − |ak| =+∞⇒ ak = ak =+∞ k=1 k=1 k=1 1which excludes ±∞ as limit. www.probability.net Tutorial 11: Complex Measures 2 Exercise 2. Let (an)n≥1 be a sequence in R, such that the series +∞ +∞ k=1 ak converges, and k=1 |ak| =+∞.LetA>0. We define: + − N = {k ≥ 1:ak ≥ 0} ,N = {k ≥ 1:ak < 0} 1. Show that N + and N − are infinite. 2. Let φ+ : N∗ → N + and φ− : N∗ → N − be two bijections. Show the existence of k1 ≥ 1 such that: k1 aφ+(k) ≥ A k=1 3. Show the existence of an increasing sequence (kp)p≥1 such that: kp aφ+(k) ≥ A k=kp−1+1 www.probability.net Tutorial 11: Complex Measures 3 for all p ≥ 1, where k0 =0. 4. Consider the permutation σ : N∗ → N∗ defined informally by: φ− ,φ+ ,...,φ+ k ,φ− ,φ+ k ,...,φ+ k ,..
    [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]
  • Appendix A. Measure and Integration
    Appendix A. Measure and integration We suppose the reader is familiar with the basic facts concerning set theory and integration as they are presented in the introductory course of analysis. In this appendix, we review them briefly, and add some more which we shall need in the text. Basic references for proofs and a detailed exposition are, e.g., [[ H a l 1 ]] , [[ J a r 1 , 2 ]] , [[ K F 1 , 2 ]] , [[ L i L ]] , [[ R u 1 ]] , or any other textbook on analysis you might prefer. A.1 Sets, mappings, relations A set is a collection of objects called elements. The symbol card X denotes the cardi- nality of the set X. The subset M consisting of the elements of X which satisfy the conditions P1(x),...,Pn(x) is usually written as M = { x ∈ X : P1(x),...,Pn(x) }.A set whose elements are certain sets is called a system or family of these sets; the family of all subsystems of a given X is denoted as 2X . The operations of union, intersection, and set difference are introduced in the standard way; the first two of these are commutative, associative, and mutually distributive. In a { } system Mα of any cardinality, the de Morgan relations , X \ Mα = (X \ Mα)and X \ Mα = (X \ Mα), α α α α are valid. Another elementary property is the following: for any family {Mn} ,whichis { } at most countable, there is a disjoint family Nn of the same cardinality such that ⊂ \ ∪ \ Nn Mn and n Nn = n Mn.Theset(M N) (N M) is called the symmetric difference of the sets M,N and denoted as M #N.
    [Show full text]
  • Measure and Integration (Under Construction)
    Measure and Integration (under construction) Gustav Holzegel∗ April 24, 2017 Abstract These notes accompany the lecture course ”Measure and Integration” at Imperial College London (Autumn 2016). They follow very closely the text “Real-Analysis” by Stein-Shakarchi, in fact most proofs are simple rephrasings of the proofs presented in the aforementioned book. Contents 1 Motivation 3 1.1 Quick review of the Riemann integral . 3 1.2 Drawbacks of the class R, motivation of the Lebesgue theory . 4 1.2.1 Limits of functions . 5 1.2.2 Lengthofcurves ....................... 5 1.2.3 TheFundamentalTheoremofCalculus . 5 1.3 Measures of sets in R ......................... 6 1.4 LiteratureandFurtherReading . 6 2 Measure Theory: Lebesgue Measure in Rd 7 2.1 Preliminaries and Notation . 7 2.2 VolumeofRectanglesandCubes . 7 2.3 Theexteriormeasure......................... 9 2.3.1 Examples ........................... 9 2.3.2 Propertiesoftheexteriormeasure . 10 2.4 Theclassof(Lebesgue)measurablesets . 12 2.4.1 The property of countable additivity . 14 2.4.2 Regularity properties of the Lebesgue measure . 15 2.4.3 Invariance properties of the Lebesgue measure . 16 2.4.4 σ-algebrasandBorelsets . 16 2.5 Constructionofanon-measurableset. 17 2.6 MeasurableFunctions ........................ 19 2.6.1 Some abstract preliminary remarks . 19 2.6.2 Definitions and equivalent formulations of measurability . 19 2.6.3 Properties 1: Behaviour under compositions . 21 2.6.4 Properties 2: Behaviour under limits . 21 2.6.5 Properties3: Behaviourof sums and products . 22 ∗Imperial College London, Department of Mathematics, South Kensington Campus, Lon- don SW7 2AZ, United Kingdom. 1 2.6.6 Thenotionof“almosteverywhere”. 22 2.7 Building blocks of integration theory .
    [Show full text]
  • Gsm076-Endmatter.Pdf
    http://dx.doi.org/10.1090/gsm/076 Measur e Theor y an d Integratio n This page intentionally left blank Measur e Theor y an d Integratio n Michael E.Taylor Graduate Studies in Mathematics Volume 76 M^^t| American Mathematical Society ^MMOT Providence, Rhode Island Editorial Board David Cox Walter Craig Nikolai Ivanov Steven G. Krantz David Saltman (Chair) 2000 Mathematics Subject Classification. Primary 28-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-76 Library of Congress Cataloging-in-Publication Data Taylor, Michael Eugene, 1946- Measure theory and integration / Michael E. Taylor. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 76) Includes bibliographical references. ISBN-13: 978-0-8218-4180-8 1. Measure theory. 2. Riemann integrals. 3. Convergence. 4. Probabilities. I. Title. II. Series. QA312.T387 2006 515/.42—dc22 2006045635 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2006 by the American Mathematical Society.
    [Show full text]
  • Dirichlet Is Natural Vincent Danos, Ilias Garnier
    Dirichlet is Natural Vincent Danos, Ilias Garnier To cite this version: Vincent Danos, Ilias Garnier. Dirichlet is Natural. MFPS 31 - Mathematical Foundations of Program- ming Semantics XXXI, Jun 2015, Nijmegen, Netherlands. pp.137-164, 10.1016/j.entcs.2015.12.010. hal-01256903 HAL Id: hal-01256903 https://hal.archives-ouvertes.fr/hal-01256903 Submitted on 15 Jan 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. MFPS 2015 Dirichlet is natural Vincent Danos1 D´epartement d'Informatique Ecole normale sup´erieure Paris, France Ilias Garnier2 School of Informatics University of Edinburgh Edinburgh, United Kingdom Abstract Giry and Lawvere's categorical treatment of probabilities, based on the probabilistic monad G, offer an elegant and hitherto unexploited treatment of higher-order probabilities. The goal of this paper is to follow this formulation to reconstruct a family of higher-order probabilities known as the Dirichlet process. This family is widely used in non-parametric Bayesian learning. Given a Polish space X, we build a family of higher-order probabilities in G(G(X)) indexed by M ∗(X) the set of non-zero finite measures over X. The construction relies on two ingredients.
    [Show full text]
  • About and Beyond the Lebesgue Decomposition of a Signed Measure
    Lecturas Matemáticas Volumen 37 (2) (2016), páginas 79-103 ISSN 0120-1980 About and beyond the Lebesgue decomposition of a signed measure Sobre el teorema de descomposición de Lebesgue de una medida con signo y un poco más Josefina Alvarez1 and Martha Guzmán-Partida2 1New Mexico State University, EE.UU. 2Universidad de Sonora, México ABSTRACT. We present a refinement of the Lebesgue decomposition of a signed measure. We also revisit the notion of mutual singularity for finite signed measures, explaining its connection with a notion of orthogonality introduced by R.C. JAMES. Key words: Lebesgue Decomposition Theorem, Mutually Singular Signed Measures, Orthogonality. RESUMEN. Presentamos un refinamiento de la descomposición de Lebesgue de una medida con signo. También revisitamos la noción de singularidad mutua para medidas con signo finitas, explicando su relación con una noción de ortogonalidad introducida por R. C. JAMES. Palabras clave: Teorema de Descomposición de Lebesgue, Medidas con Signo Mutua- mente Singulares, Ortogonalidad. 2010 AMS Mathematics Subject Classification. Primary 28A10; Secondary 28A33, Third 46C50. 1. Introduction The main goal of this expository article is to present a refinement, not often found in the literature, of the Lebesgue decomposition of a signed measure. For the sake of completeness, we begin with a collection of preliminary definitions and results, including many comments, examples and counterexamples. We continue with a section dedicated 80 Josefina Alvarez et al. About and beyond the Lebesgue decomposition... specifically to the Lebesgue decomposition of a signed measure, where we also present a short account of its historical development. Next comes the centerpiece of our exposition, a refinement of the Lebesgue decomposition of a signed measure, which we prove in detail.
    [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]
  • (Measure Theory for Dummies) UWEE Technical Report Number UWEETR-2006-0008
    A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UWEE Technical Report Number UWEETR-2006-0008 May 2006 Department of Electrical Engineering University of Washington Box 352500 Seattle, Washington 98195-2500 PHN: (206) 543-2150 FAX: (206) 543-3842 URL: http://www.ee.washington.edu A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 University of Washington, Dept. of EE, UWEETR-2006-0008 May 2006 Abstract This tutorial is an informal introduction to measure theory for people who are interested in reading papers that use measure theory. The tutorial assumes one has had at least a year of college-level calculus, some graduate level exposure to random processes, and familiarity with terms like “closed” and “open.” The focus is on the terms and ideas relevant to applied probability and information theory. There are no proofs and no exercises. Measure theory is a bit like grammar, many people communicate clearly without worrying about all the details, but the details do exist and for good reasons. There are a number of great texts that do measure theory justice. This is not one of them. Rather this is a hack way to get the basic ideas down so you can read through research papers and follow what’s going on. Hopefully, you’ll get curious and excited enough about the details to check out some of the references for a deeper understanding.
    [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]