On Stochastic Distributions and Currents
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Integral Representation of a Linear Functional on Function Spaces
INTEGRAL REPRESENTATION OF LINEAR FUNCTIONALS ON FUNCTION SPACES MEHDI GHASEMI Abstract. Let A be a vector space of real valued functions on a non-empty set X and L : A −! R a linear functional. Given a σ-algebra A, of subsets of X, we present a necessary condition for L to be representable as an integral with respect to a measure µ on X such that elements of A are µ-measurable. This general result then is applied to the case where X carries a topological structure and A is a family of continuous functions and naturally A is the Borel structure of X. As an application, short solutions for the full and truncated K-moment problem are presented. An analogue of Riesz{Markov{Kakutani representation theorem is given where Cc(X) is replaced with whole C(X). Then we consider the case where A only consists of bounded functions and hence is equipped with sup-norm. 1. Introduction A positive linear functional on a function space A ⊆ RX is a linear map L : A −! R which assigns a non-negative real number to every function f 2 A that is globally non-negative over X. The celebrated Riesz{Markov{Kakutani repre- sentation theorem states that every positive functional on the space of continuous compactly supported functions over a locally compact Hausdorff space X, admits an integral representation with respect to a regular Borel measure on X. In symbols R L(f) = X f dµ, for all f 2 Cc(X). Riesz's original result [12] was proved in 1909, for the unit interval [0; 1]. -
Geometric Integration Theory Contents
Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants . -
Invariant Radon Measures on Measured Lamination Space
INVARIANT RADON MEASURES ON MEASURED LAMINATION SPACE URSULA HAMENSTADT¨ Abstract. Let S be an oriented surface of genus g ≥ 0 with m ≥ 0 punctures and 3g − 3+ m ≥ 2. We classify all Radon measures on the space of measured geodesic laminations which are invariant under the action of the mapping class group of S. 1. Introduction Let S be an oriented surface of finite type, i.e. S is a closed surface of genus g 0 from which m 0 points, so-called punctures, have been deleted. We assume that≥ 3g 3 + m 1,≥ i.e. that S is not a sphere with at most 3 punctures or a torus without− puncture.≥ In particular, the Euler characteristic of S is negative. Then the Teichm¨uller space (S) of S is the quotient of the space of all complete hyperbolic metrics of finite volumeT on S under the action of the group of diffeomorphisms of S which are isotopic to the identity. The mapping class group MCG(S) of all isotopy classes of orientation preserving diffeomorphisms of S acts properly discontinuously on (S) with quotient the moduli space Mod(S). T A geodesic lamination for a fixed choice of a complete hyperbolic metric of finite volume on S is a compact subset of S foliated into simple geodesics. A measured geodesic lamination is a geodesic lamination together with a transverse translation invariant measure. The space of all measured geodesic laminations on S, equipped with the weak∗-topology,ML is homeomorphic to S6g−7+2m (0, ) where S6g−7+2m is the 6g 7+2m-dimensional sphere. -
SOME ALGEBRAIC DEFINITIONS and CONSTRUCTIONS Definition
SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS Definition 1. A monoid is a set M with an element e and an associative multipli- cation M M M for which e is a two-sided identity element: em = m = me for all m M×. A−→group is a monoid in which each element m has an inverse element m−1, so∈ that mm−1 = e = m−1m. A homomorphism f : M N of monoids is a function f such that f(mn) = −→ f(m)f(n) and f(eM )= eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n M, we often write the product as +, the identity element as 0, and the inverse of∈m as m. As a convention, it is convenient to say that a commutative monoid is “Abelian”− when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1. Definition 2. The Grothendieck construction on an Abelian monoid is an Abelian group G(M) together with a homomorphism of Abelian monoids i : M G(M) such that, for any Abelian group A and homomorphism of Abelian monoids−→ f : M A, there exists a unique homomorphism of Abelian groups f˜ : G(M) A −→ −→ such that f˜ i = f. ◦ We construct G(M) explicitly by taking equivalence classes of ordered pairs (m,n) of elements of M, thought of as “m n”, under the equivalence relation generated by (m,n) (m′,n′) if m + n′ = −n + m′. -
MTH 620: 2020-04-28 Lecture
MTH 620: 2020-04-28 lecture Alexandru Chirvasitu Introduction In this (last) lecture we'll mix it up a little and do some topology along with the homological algebra. I wanted to bring up the cohomology of profinite groups if only briefly, because we discussed these in MTH619 in the context of Galois theory. Consider this as us connecting back to that material. 1 Topological and profinite groups This is about the cohomology of profinite groups; we discussed these briefly in MTH619 during Fall 2019, so this will be a quick recollection. First things first though: Definition 1.1 A topological group is a group G equipped with a topology, such that both the multiplication G × G ! G and the inverse map g 7! g−1 are continuous. Unless specified otherwise, all of our topological groups will be assumed separated (i.e. Haus- dorff) as topological spaces. Ordinary, plain old groups can be regarded as topological groups equipped with the discrete topology (i.e. so that all subsets are open). When I want to be clear we're considering plain groups I might emphasize that by referring to them as `discrete groups'. The main concepts we will work with are covered by the following two definitions (or so). Definition 1.2 A profinite group is a compact (and as always for us, Hausdorff) topological group satisfying any of the following equivalent conditions: Q G embeds as a closed subgroup into a product i2I Gi of finite groups Gi, equipped with the usual product topology; For every open neighborhood U of the identity 1 2 G there is a normal, open subgroup N E G contained in U. -
Arxiv:2011.03427V2 [Math.AT] 12 Feb 2021 Riae Fmp Ffiiest.W Eosrt Hscategor This Demonstrate We Sets
HYPEROCTAHEDRAL HOMOLOGY FOR INVOLUTIVE ALGEBRAS DANIEL GRAVES Abstract. Hyperoctahedral homology is the homology theory associated to the hyperoctahe- dral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyperoctahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the group. Introduction Hyperoctahedral homology for involutive algebras was introduced by Fiedorowicz [Fie, Sec- tion 2]. It is the homology theory associated to the hyperoctahedral crossed simplicial group [FL91, Section 3]. Fiedorowicz and Loday [FL91, 6.16] had shown that the homology theory constructed from the hyperoctahedral crossed simplicial group via a contravariant bar construc- tion, analogously to cyclic homology, was isomorphic to Hochschild homology and therefore did not detect the action of the hyperoctahedral groups. Fiedorowicz demonstrated that a covariant bar construction did detect this action and sketched results connecting the hyperoctahedral ho- mology of monoid algebras and group algebras to May’s two-sided bar construction and infinite loop spaces, though these were never published. In Section 1 we recall the hyperoctahedral groups. We recall the hyperoctahedral category ∆H associated to the hyperoctahedral crossed simplicial group. This category encodes an involution compatible with an order-preserving multiplication. We introduce the category of involutive non-commutative sets, which encodes the same information by adding data to the preimages of maps of finite sets. -
10 Heat Equation: Interpretation of the Solution
Math 124A { October 26, 2011 «Viktor Grigoryan 10 Heat equation: interpretation of the solution Last time we considered the IVP for the heat equation on the whole line u − ku = 0 (−∞ < x < 1; 0 < t < 1); t xx (1) u(x; 0) = φ(x); and derived the solution formula Z 1 u(x; t) = S(x − y; t)φ(y) dy; for t > 0; (2) −∞ where S(x; t) is the heat kernel, 1 2 S(x; t) = p e−x =4kt: (3) 4πkt Substituting this expression into (2), we can rewrite the solution as 1 1 Z 2 u(x; t) = p e−(x−y) =4ktφ(y) dy; for t > 0: (4) 4πkt −∞ Recall that to derive the solution formula we first considered the heat IVP with the following particular initial data 1; x > 0; Q(x; 0) = H(x) = (5) 0; x < 0: Then using dilation invariance of the Heaviside step function H(x), and the uniquenessp of solutions to the heat IVP on the whole line, we deduced that Q depends only on the ratio x= t, which lead to a reduction of the heat equation to an ODE. Solving the ODE and checking the initial condition (5), we arrived at the following explicit solution p x= 4kt 1 1 Z 2 Q(x; t) = + p e−p dp; for t > 0: (6) 2 π 0 The heat kernel S(x; t) was then defined as the spatial derivative of this particular solution Q(x; t), i.e. @Q S(x; t) = (x; t); (7) @x and hence it also solves the heat equation by the differentiation property. -
Chapter 3 Proofs
Chapter 3 Proofs Many mathematical proofs use a small range of standard outlines: direct proof, examples/counter-examples, and proof by contrapositive. These notes explain these basic proof methods, as well as how to use definitions of new concepts in proofs. More advanced methods (e.g. proof by induction, proof by contradiction) will be covered later. 3.1 Proving a universal statement Now, let’s consider how to prove a claim like For every rational number q, 2q is rational. First, we need to define what we mean by “rational”. A real number r is rational if there are integers m and n, n = 0, such m that r = n . m In this definition, notice that the fraction n does not need to satisfy conditions like being proper or in lowest terms So, for example, zero is rational 0 because it can be written as 1 . However, it’s critical that the two numbers 27 CHAPTER 3. PROOFS 28 in the fraction be integers, since even irrational numbers can be written as π fractions with non-integers on the top and/or bottom. E.g. π = 1 . The simplest technique for proving a claim of the form ∀x ∈ A, P (x) is to pick some representative value for x.1. Think about sticking your hand into the set A with your eyes closed and pulling out some random element. You use the fact that x is an element of A to show that P (x) is true. Here’s what it looks like for our example: Proof: Let q be any rational number. -
Structure Theory of Set Addition
Structure Theory of Set Addition Notes by B. J. Green1 ICMS Instructional Conference in Combinatorial Aspects of Mathematical Analysis, Edinburgh March 25 { April 5 2002. 1 Lecture 1: Pl¨unnecke's Inequalities 1.1 Introduction The object of these notes is to explain a recent proof by Ruzsa of a famous result of Freiman, some significant modifications of Ruzsa's proof due to Chang, and all the background material necessary to understand these arguments. Freiman's theorem concerns the structure of sets with small sumset. Let A be a subset of an abelian group G, and define the sumset A + A to be the set of all pairwise sums a + a0, where a; a0 are (not necessarily distinct) elements of A. If A = n then A + A n, and equality can occur (for example if A is a subgroup of G).j Inj the otherj directionj ≥ we have A + A n(n + 1)=2, and equality can occur here too, for example when G = Z and j j ≤ 2 n 1 A = 1; 3; 3 ;:::; 3 − . It is easy to construct similar examples of sets with large sumset, but ratherf harder to findg examples with A + A small. Let us think more carefully about this problem in the special case G = Z. Proposition 1 Let A Z have size n. Then A + A 2n 1, with equality if and only if A is an arithmetic progression⊆ of length n. j j ≥ − Proof. Write A = a1; : : : ; an where a1 < a2 < < an. Then we have f g ··· a1 + a1 < a1 + a2 < < a1 + an < a2 + an < < an + an; ··· ··· which amounts to an exhibition of 2n 1 distinct elements of A + A. -
Nonlinear Structures Determined by Measures on Banach Spaces Mémoires De La S
MÉMOIRES DE LA S. M. F. K. DAVID ELWORTHY Nonlinear structures determined by measures on Banach spaces Mémoires de la S. M. F., tome 46 (1976), p. 121-130 <http://www.numdam.org/item?id=MSMF_1976__46__121_0> © Mémoires de la S. M. F., 1976, tous droits réservés. L’accès aux archives de la revue « Mémoires de la S. M. F. » (http://smf. emath.fr/Publications/Memoires/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Journees Geom. dimens. infinie [1975 - LYON ] 121 Bull. Soc. math. France, Memoire 46, 1976, p. 121 - 130. NONLINEAR STRUCTURES DETERMINED BY MEASURES ON BANACH SPACES By K. David ELWORTHY 0. INTRODUCTION. A. A Gaussian measure y on a separable Banach space E, together with the topolcT- gical vector space structure of E, determines a continuous linear injection i : H -> E, of a Hilbert space H, such that y is induced by the canonical cylinder set measure of H. Although the image of H has measure zero, nevertheless H plays a dominant role in both linear and nonlinear analysis involving y, [ 8] , [9], [10] . The most direct approach to obtaining measures on a Banach manifold M, related to its differential structure, requires a lot of extra structure on the manifold : for example a linear map i : H -> T M for each x in M, and even a subset M-^ of M which has the structure of a Hilbert manifold, [6] , [7]. -
A Direct Approach to Plateau's Problem 1
A DIRECT APPROACH TO PLATEAU'S PROBLEM C. DE LELLIS, F. GHIRALDIN, AND F. MAGGI Abstract. We provide a compactness principle which is applicable to different formulations of Plateau's problem in codimension 1 and which is exclusively based on the theory of Radon measures and elementary comparison arguments. Exploiting some additional techniques in geometric measure theory, we can use this principle to give a different proof of a theorem by Harrison and Pugh and to answer a question raised by Guy David. 1. Introduction Since the pioneering work of Reifenberg there has been an ongoing interest into formulations of Plateau's problem involving the minimization of the Hausdorff measure on closed sets coupled with some notion of \spanning a given boundary". More precisely consider any closed set H ⊂ n+1 n+1 R and assume to have a class P(H) of relatively closed subsets K of R nH, which encodes a particular notion of \K bounds H". Correspondingly there is a formulation of Plateau's problem, namely the minimum for such problem is n m0 := inffH (K): K 2 P(H)g ; (1.1) n and a minimizing sequence fKjg ⊂ P(H) is characterized by the property H (Kj) ! m0. Two good motivations for considering this kind of approach rather than the one based on integer rectifiable currents are that, first, not every interesting boundary can be realized as an integer 3 rectifiable cycle and, second, area minimizing 2-d currents in R are always smooth away from their boundaries, in contrast to what one observes with real world soap films. -
Harmonic Analysis Meets Geometric Measure Theory
Geometry of Measures: Harmonic Analysis meets Geometric Measure Theory T. Toro ∗ 1 Introduction One of the central questions in Geometric Measure Theory is the extend to which the regularity of a measure determines the geometry of its support. This type of question was initially studied by Besicovitch, and then pursued by many authors among others Marstrand, Mattila and Preiss. These authors focused their attention on the extend to which the behavior of the density ratio of a m given Radon measure µ in R with respect to Hausdorff measure determines the regularity of the µ(B(x;r)) m support of the measure, i.e. what information does the quantity rs for x 2 R , r > 0 and s > 0 encode. In the context of the study of minimizers of area, perimeter and similar functionals the correct density ratio is monotone, the density exists and its behavior is the key to study the regularity and structure of these minimizers. This question has also been studied in the context of Complex Analysis. In this case, the Radon measure of interest is the harmonic measure. The general question is to what extend the structure of the boundary of a domain can be fully understood in terms of the behavior of the harmonic measure. Some of the first authors to study this question were Carleson, Jones, Wolff and Makarov. The goal of this paper is to present two very different type of problems, and the unifying techniques that lead to a resolution of both. In Section 2 we look at the history of the question initially studied by Besicovitch in dimension 2, as well as some of its offsprings.