DOCSLIB.ORG

HARMONIC ANALYSIS 1. Maximal Function for a Locally Integrable

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
HARMONIC ANALYSIS 1. Maximal Function for a Locally Integrable HARMONIC ANALYSIS PIOTR HAJLASZ 1. Maximal function 1 n For a locally integrable function f 2 Lloc(R ) the Hardy-Littlewood max- imal function is defined by Z n Mf(x) = sup jf(y)j dy; x 2 R : r>0 B(x;r) The operator M is not linear but it is subadditive. We say that an operator T from a space of measurable functions into a space of measurable functions is subadditive if jT (f1 + f2)(x)j ≤ jT f1(x)j + jT f2(x)j a.e. and jT (kf)(x)j = jkjjT f(x)j for k 2 C. The following integrability result, known also as the maximal theorem, plays a fundamental role in many areas of mathematical analysis. p n Theorem 1.1 (Hardy-Littlewood-Wiener). If f 2 L (R ), 1 ≤ p ≤ 1, then Mf < 1 a.e. Moreover 1 n (a) For f 2 L (R ) 5n Z (1.1) jfx : Mf(x) > tgj ≤ jfj for all t > 0. t Rn p n p n (b) If f 2 L (R ), 1 < p ≤ 1, then Mf 2 L (R ) and p 1=p kMfk ≤ 2 · 5n=p kfk for 1 < p < 1, p p − 1 p kMfk1 ≤ kfk1 : Date: March 28, 2012. 1 2 PIOTR HAJLASZ The estimate (1.1) is called weak type estimate. 1 n 1 n Note that if f 2 L (R ) is a nonzero function, then Mf 62 L (R ). Indeed, R if λ = B(0;R) jfj > 0, then for jxj > R we have Z λ Mf(x) ≥ jfj ≥ n ; B(x;R+jxj) !n(R + jxj) n and the function on the right hand side is not integrable on R . Thus the statement (b) of the theorem is not true for p = 1. 1 n If g 2 L (R ), then the Chebyschev inequality 1 Z jfx : jg(x)j > tgj ≤ jgj for t > 0 t Rn is easy to prove. Hence the inequality at (a) would follow from boundedness of of Mf in L1. Unfortunately Mf is not integrable and (a) is the best what we can get for p = 1. Before we prove the theorem we will show that it implies the Lebesgue differentiation theorem. 1 n Theorem 1.2 (Lebesgue differentiation theorem). If f 2 Lloc(R ), then Z lim f(y) dy = f(x) a.e. r!0 B(x;r) 1 n Proof. Since the theorem is local in nature we can assume that f 2 L (R ). R Let fr(x) = B(x;r)f(y) dy and define Ωf(x) = lim sup fr(x) − lim inf fr(x) : r!0 r!0 1 It suffices to prove that Ωf = 0 a.e. and that fr ! f in L . Indeed, the first property means that fr converges a.e. to a measurable function g while the second one implies that for a subsequence fri ! f a.e. and hence g = f a.e. Observe that Ωf ≤ 2Mf and hence for any " > 0 Theorem 1.1(a) yields C Z jfx :Ωf(x) > "gj ≤ jfj : " Rn 2 Let h be a continuous function such that kf − hk1 < " . Continuity of h implies Ωh = 0 everywhere and hence Ωf ≤ Ω(f − h) + Ωh = Ω(f − h) ; so C Z jfΩf > "gj ≤ jfΩ(f − h) > "gj ≤ jf − hj ≤ C": " Rn HARMONIC ANALYSIS 3 Since " > 0 can be arbitrarily small we conclude Ωf = 0 a.e. We are left 1 with the proof that fr ! f in L . We have Z Z Z jfr(x) − f(x)j dx ≤ jf(y) − f(x)j dy dx Rn Rn B(x;r) Z Z = jf(x + y) − f(x)j dy dx Rn B(0;r) Z (1.2) = kfy − fk1 dy ; B(0;r) 1 where fy(x) = f(x + y). Since fy ! f in L as y ! 0 the right hand side of (1.2) converges to 0 as r ! 0. 2 1 n If f 2 Lloc(R ), then we can define f at every point by the formula Z (1.3) f(x) := lim sup f(y) dy : r!0 B(x;r) According to the Lebesgue differentiation theorem this is a representative of f in the class of functions that coincide with f a.e. 1 n n Definition. Let f 2 Lloc(R ). We say that x 2 R is a Lebesgue point of f if Z lim jf(y) − f(x)j dy = 0 ; r!0 B(x;r) where f(x) is defined by (1.3). 1 n Theorem 1.3. If f 2 Lloc(R ), then the set of points that are not Lebesgue points of f has measure zero. Proof. For c 2 Q let Ec be the set of points for which Z (1.4) lim jf(y) − cj dy = jf(x) − cj r!0 B(x;r) does not hold. Clearly jE j = 0 and hence the set E = S E has measure c c2Q c n n zero. Thus for x 2 R n E and all c 2 Q, (1.4) is satisfied. If x 2 R n E and f(x) 2 R, approximating f(x) by rational numbers one can easily check that Z lim jf(y) − f(x)j dy = jf(x) − f(x)j = 0 : r!0 B(x;r) The proof is complete. 2 n We can generalize the above result as follows. We say that x 2 R is a p p-Lebesgue point of f 2 Lloc, 1 ≤ p < 1 if Z lim jf(y) − f(x)jp dy = 0 : r!0 B(x;r) The same method as the one used above leads to the following result that we leave an an exercise. 4 PIOTR HAJLASZ p n Theorem 1.4. If f 2 Lloc(R ), 1 ≤ p < 1, then the set of points that are not p-Lebesgue points of f has measure zero. n n Definition. Let E ⊂ R be a measurable set. We say that x 2 R is a density point of E if jB(x; r) \ Ej lim = 1 : r!0 jB(x; r)j Applying the Lebesgue theorem to f = χE we obtain n Theorem 1.5. Almost every point of a measurable set E ⊂ R is its density n point and a.e. point of R n E is not a density point of E. In the Lebesgue theorem we have seen that the averages of f over balls converge to f(x) and it is natural to inquire if we can replace balls by other sets like cubes or even balls, but not centered at x. n Definition. We say that a family F of measurable subsets of R is regular n at x 2 R if (a) The sets are bounded and have positive measure; (b) There is a sequence Sn 2 F with jSnj ! 0; (c) There is a constant C > 0 such that every S 2 F is contained in a ball B ⊃ S centered at x such that jSj ≥ CjBj. n Example. The family of all cubes Q in R such that the distance of Q to x is no more than Cdiam Q is regular. 1 n Theorem 1.6. If f 2 Lloc(R ), x is a Lebesgue point of f, and F is regular at x, then Z lim f(y) dy = f(x) : S2F jSj!0 S Proof. For S 2 F denote by rS the radius of a ball BS = B(x; rS) such that S ⊂ BS and jSj ≥ CjBSj. Clearly if jSj ! 0, then rS ! 0. We have Z Z f(y) dy − f(x) ≤ jf(y) − f(x)j dy S S Z ≤ jSj−1 jf(y) − f(x)j dy BS Z ≤ C−1 jf(y) − f(x)j dy ! 0 BS as jSj ! 0. 2 HARMONIC ANALYSIS 5 Note that if F is a regular family at 0 and we define the maximal function associated with F by Z MF f(x) = sup jf(x − y)j dy ; S2F S then it is a routine calculation to show that (1.5) MF f(x) ≤ CMf(x) ; so MF satisfies the claim of Theorem 1.1 (with different constants). In p particular MF is a bounded operator in L , 1 < p ≤ 1. In the proof of Theorem 1.1 we will need the following two results. Theorem 1.7 (Cavalieri's principle). If µ is a σ-finite measure on X and Φ : [0; 1) ! [0; 1) is increasing, absolutely continuous and Φ(0) = 0, then Z Z 1 Φ(jfj) dµ = Φ0(t)µ(fjfj > tg) dt : X 0 Proof. The result follows immediately from the equality Z Z Z jf(x)j Φ(jf(x)j) dµ(x) = Φ0(t) dt dµ(x) X X 0 and the Fubini theorem. 2 Corollary 1.8. If µ is a σ-finite measure on X and 0 < p < 1, then Z Z 1 jfjp dµ = p tp−1µ(fjfj > tg) dt : X 0 The next result has many applications that go beyond the maximal the- orem. Theorem 1.9 (5r-covering lemma). Let B be a family of balls in a metric space such that supfdiam B : B 2 Bg < 1. Then there is a subfamily of pairwise disjoint balls B0 ⊂ B such that [ [ B ⊂ 5B: B2B B2B0 If the metric space is separable, then the family B0 is countable and we can 0 1 arrange it as a sequence B = fBigi=1, so 1 [ [ B ⊂ 5Bi : B2B i=1 Remark. Here B can be either a family of open balls or closed balls.
Load more

Recommended publications
  • A Convenient Category for Higher-Order Probability Theory
    A Convenient Category for Higher-Order Probability Theory Chris Heunen Ohad Kammar Sam Staton Hongseok Yang University of Edinburgh, UK University of Oxford, UK University of Oxford, UK University of Oxford, UK Abstract—Higher-order probabilistic programming languages 1 (defquery Bayesian-linear-regression allow programmers to write sophisticated models in machine 2 let let sample normal learning and statistics in a succinct and structured way, but step ( [f ( [s ( ( 0.0 3.0)) 3 sample normal outside the standard measure-theoretic formalization of proba- b ( ( 0.0 3.0))] 4 fn + * bility theory. Programs may use both higher-order functions and ( [x] ( ( s x) b)))] continuous distributions, or even define a probability distribution 5 (observe (normal (f 1.0) 0.5) 2.5) on functions. But standard probability theory does not handle 6 (observe (normal (f 2.0) 0.5) 3.8) higher-order functions well: the category of measurable spaces 7 (observe (normal (f 3.0) 0.5) 4.5) is not cartesian closed. 8 (observe (normal (f 4.0) 0.5) 6.2) Here we introduce quasi-Borel spaces. We show that these 9 (observe (normal (f 5.0) 0.5) 8.0) spaces: form a new formalization of probability theory replacing 10 (predict :f f))) measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and proba- bility by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti’s theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.
    [Show full text]
  • Jointly Measurable and Progressively Measurable Stochastic Processes
    Jointly measurable and progressively measurable stochastic processes Jordan Bell [email protected] Department of Mathematics, University of Toronto June 18, 2015 1 Jointly measurable stochastic processes d Let E = R with Borel E , let I = R≥0, which is a topological space with the subspace topology inherited from R, and let (Ω; F ;P ) be a probability space. For a stochastic process (Xt)t2I with state space E, we say that X is jointly measurable if the map (t; !) 7! Xt(!) is measurable BI ⊗ F ! E . For ! 2 Ω, the path t 7! Xt(!) is called left-continuous if for each t 2 I, Xs(!) ! Xt(!); s " t: We prove that if the paths of a stochastic process are left-continuous then the stochastic process is jointly measurable.1 Theorem 1. If X is a stochastic process with state space E and all the paths of X are left-continuous, then X is jointly measurable. Proof. For n ≥ 1, t 2 I, and ! 2 Ω, let 1 n X Xt (!) = 1[k2−n;(k+1)2−n)(t)Xk2−n (!): k=0 n Each X is measurable BI ⊗ F ! E : for B 2 E , 1 n [ −n −n f(t; !) 2 I × Ω: Xt (!) 2 Bg = [k2 ; (k + 1)2 ) × fXk2−n 2 Bg: k=0 −n −n Let t 2 I. For each n there is a unique kn for which t 2 [kn2 ; (kn +1)2 ), and n −n −n thus Xt (!) = Xkn2 (!). Furthermore, kn2 " t, and because s 7! Xs(!) is n −n left-continuous, Xkn2 (!) ! Xt(!).
    [Show full text]
  • Shape Analysis, Lebesgue Integration and Absolute Continuity Connections
    NISTIR 8217 Shape Analysis, Lebesgue Integration and Absolute Continuity Connections Javier Bernal This publication is available free of charge from: https://doi.org/10.6028/NIST.IR.8217 NISTIR 8217 Shape Analysis, Lebesgue Integration and Absolute Continuity Connections Javier Bernal Applied and Computational Mathematics Division Information Technology Laboratory This publication is available free of charge from: https://doi.org/10.6028/NIST.IR.8217 July 2018 INCLUDES UPDATES AS OF 07-18-2018; SEE APPENDIX U.S. Department of Commerce Wilbur L. Ross, Jr., Secretary National Institute of Standards and Technology Walter Copan, NIST Director and Undersecretary of Commerce for Standards and Technology ______________________________________________________________________________________________________ This Shape Analysis, Lebesgue Integration and publication Absolute Continuity Connections Javier Bernal is National Institute of Standards and Technology, available Gaithersburg, MD 20899, USA free of Abstract charge As shape analysis of the form presented in Srivastava and Klassen’s textbook “Functional and Shape Data Analysis” is intricately related to Lebesgue integration and absolute continuity, it is advantageous from: to have a good grasp of the latter two notions. Accordingly, in these notes we review basic concepts and results about Lebesgue integration https://doi.org/10.6028/NIST.IR.8217 and absolute continuity. In particular, we review fundamental results connecting them to each other and to the kind of shape analysis, or more generally, functional data analysis presented in the aforeme- tioned textbook, in the process shedding light on important aspects of all three notions. Many well-known results, especially most results about Lebesgue integration and some results about absolute conti- nuity, are presented without proofs.
    [Show full text]
  • Three Problems of the Theory of Choice on Random Sets
    DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 91125 THREE PROBLEMS OF THE THEORY OF CHOICE ON RANDOM SETS B. A. Berezovskiy,* Yu. M. Baryshnikov, A. Gnedin V. Institute of Control Sciences Academy of Sciences of the U.S.S.R, Moscow *Guest of the California Institute of Technology SOCIAL SCIENCE WORKING PAPER 661 December 1987 THREE PROBLEMS OF THE THEORY OF CHOICE ON RANDOM SETS B. A. Berezovskiy, Yu. M. Baryshnikov, A. V. Gnedin Institute of Control Sciences Academy of Sciences of the U.S.S.R., Moscow ABSTRACT This paper discusses three problems which are united not only by the common topic of research stated in thetitle, but also by a somewhat surprising interlacing of the methods and techniques used. In the first problem, an attempt is made to resolve a very unpleasant metaproblem arising in general choice theory: why theconditions of rationality are not really necessary or, in other words, why in every-day life we are quite satisfied with choice methods which are far from being ideal. The answer, substantiated by a number of results, is as follows: situations in which the choice function "misbehaves" are very seldom met in large presentations. the second problem, an overview of our studies is given on the problem of statistical propertiesIn of choice. One of themost astonishing phenomenon found when we deviate from scalar­ extremal choice functions is in stable multiplicity of choice. If our presentation is random, then a random number of alternatives is chosen in it. But how many? The answer isn't trival, and may be sought in many differentdirections.
    [Show full text]
  • LEBESGUE MEASURE and L2 SPACE. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L2 Space 4 Acknowledgments 9 References
    LEBESGUE MEASURE AND L2 SPACE. ANNIE WANG Abstract. This paper begins with an introduction to measure spaces and the Lebesgue theory of measure and integration. Several important theorems regarding the Lebesgue integral are then developed. Finally, we prove the completeness of the L2(µ) space and show that it is a metric space, and a Hilbert space. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L2 Space 4 Acknowledgments 9 References 9 1. Measure Spaces Definition 1.1. Suppose X is a set. Then X is said to be a measure space if there exists a σ-ring M (that is, M is a nonempty family of subsets of X closed under countable unions and under complements)of subsets of X and a non-negative countably additive set function µ (called a measure) defined on M . If X 2 M, then X is said to be a measurable space. For example, let X = Rp, M the collection of Lebesgue-measurable subsets of Rp, and µ the Lebesgue measure. Another measure space can be found by taking X to be the set of all positive integers, M the collection of all subsets of X, and µ(E) the number of elements of E. We will be interested only in a special case of the measure, the Lebesgue measure. The Lebesgue measure allows us to extend the notions of length and volume to more complicated sets. Definition 1.2. Let Rp be a p-dimensional Euclidean space . We denote an interval p of R by the set of points x = (x1; :::; xp) such that (1.3) ai ≤ xi ≤ bi (i = 1; : : : ; p) Definition 1.4.
    [Show full text]
  • 1 Measurable Functions
    36-752 Advanced Probability Overview Spring 2018 2. Measurable Functions, Random Variables, and Integration Instructor: Alessandro Rinaldo Associated reading: Sec 1.5 of Ash and Dol´eans-Dade; Sec 1.3 and 1.4 of Durrett. 1 Measurable Functions 1.1 Measurable functions Measurable functions are functions that we can integrate with respect to measures in much the same way that continuous functions can be integrated \dx". Recall that the Riemann integral of a continuous function f over a bounded interval is defined as a limit of sums of lengths of subintervals times values of f on the subintervals. The measure of a set generalizes the length while elements of the σ-field generalize the intervals. Recall that a real-valued function is continuous if and only if the inverse image of every open set is open. This generalizes to the inverse image of every measurable set being measurable. Definition 1 (Measurable Functions). Let (Ω; F) and (S; A) be measurable spaces. Let f :Ω ! S be a function that satisfies f −1(A) 2 F for each A 2 A. Then we say that f is F=A-measurable. If the σ-field’s are to be understood from context, we simply say that f is measurable. Example 2. Let F = 2Ω. Then every function from Ω to a set S is measurable no matter what A is. Example 3. Let A = f?;Sg. Then every function from a set Ω to S is measurable, no matter what F is. Proving that a function is measurable is facilitated by noticing that inverse image commutes with union, complement, and intersection.
    [Show full text]
  • Notes 2 : Measure-Theoretic Foundations II
    Notes 2 : Measure-theoretic foundations II Math 733-734: Theory of Probability Lecturer: Sebastien Roch References: [Wil91, Chapters 4-6, 8], [Dur10, Sections 1.4-1.7, 2.1]. 1 Independence 1.1 Definition of independence Let (Ω; F; P) be a probability space. DEF 2.1 (Independence) Sub-σ-algebras G1; G2;::: of F are independent for all Gi 2 Gi, i ≥ 1, and distinct i1; : : : ; in we have n Y P[Gi1 \···\ Gin ] = P[Gij ]: j=1 Specializing to events and random variables: DEF 2.2 (Independent RVs) RVs X1;X2;::: are independent if the σ-algebras σ(X1); σ(X2);::: are independent. DEF 2.3 (Independent Events) Events E1;E2;::: are independent if the σ-algebras c Ei = f;;Ei;Ei ; Ωg; i ≥ 1; are independent. The more familiar definitions are the following: THM 2.4 (Independent RVs: Familiar definition) RVs X, Y are independent if and only if for all x; y 2 R P[X ≤ x; Y ≤ y] = P[X ≤ x]P[Y ≤ y]: THM 2.5 (Independent events: Familiar definition) Events E1, E2 are indepen- dent if and only if P[E1 \ E2] = P[E1]P[E2]: 1 Lecture 2: Measure-theoretic foundations II 2 The proofs of these characterizations follows immediately from the following lemma. LEM 2.6 (Independence and π-systems) Suppose that G and H are sub-σ-algebras and that I and J are π-systems such that σ(I) = G; σ(J ) = H: Then G and H are independent if and only if I and J are, i.e., P[I \ J] = P[I]P[J]; 8I 2 I;J 2 J : Proof: Suppose I and J are independent.
    [Show full text]
  • The Halász-Székely Barycenter
    THE HALÁSZ–SZÉKELY BARYCENTER JAIRO BOCHI, GODOFREDO IOMMI, AND MARIO PONCE Abstract. We introduce a notion of barycenter of a probability measure re- lated to the symmetric mean of a collection of nonnegative real numbers. Our definition is inspired by the work of Halász and Székely, who in 1976 proved a law of large numbers for symmetric means. We study analytic properties of this Halász–Székely barycenter. We establish fundamental inequalities that relate the symmetric mean of a list of nonnegative real numbers with the barycenter of the measure uniformly supported on these points. As consequence, we go on to establish an ergodic theorem stating that the symmetric means of a se- quence of dynamical observations converges to the Halász–Székely barycenter of the corresponding distribution. 1. Introduction Means have fascinated man for a long time. Ancient Greeks knew the arith- metic, geometric, and harmonic means of two positive numbers (which they may have learned from the Babylonians); they also studied other types of means that can be defined using proportions: see [He, pp. 85–89]. Newton and Maclaurin en- countered the symmetric means (more about them later). Huygens introduced the notion of expected value and Jacob Bernoulli proved the first rigorous version of the law of large numbers: see [Mai, pp. 51, 73]. Gauss and Lagrange exploited the connection between the arithmetico-geometric mean and elliptic functions: see [BB]. Kolmogorov and other authors considered means from an axiomatic point of view and determined when a mean is arithmetic under a change of coordinates (i.e. quasiarithmetic): see [HLP, p.
    [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]
  • TOPIC 0 Introduction
    TOPIC 0 Introduction 1 Review Course: Math 535 (http://www.math.wisc.edu/~roch/mmids/) - Mathematical Methods in Data Science (MMiDS) Author: Sebastien Roch (http://www.math.wisc.edu/~roch/), Department of Mathematics, University of Wisconsin-Madison Updated: Sep 21, 2020 Copyright: © 2020 Sebastien Roch We first review a few basic concepts. 1.1 Vectors and norms For a vector ⎡ 푥 ⎤ ⎢ 1 ⎥ 푥 ⎢ 2 ⎥ 푑 퐱 = ⎢ ⎥ ∈ ℝ ⎢ ⋮ ⎥ ⎣ 푥푑 ⎦ the Euclidean norm of 퐱 is defined as ‾‾푑‾‾‾‾ ‖퐱‖ = 푥2 = √퐱‾‾푇‾퐱‾ = ‾⟨퐱‾‾, ‾퐱‾⟩ 2 ∑ 푖 √ ⎷푖=1 푇 where 퐱 denotes the transpose of 퐱 (seen as a single-column matrix) and 푑 ⟨퐮, 퐯⟩ = 푢 푣 ∑ 푖 푖 푖=1 is the inner product (https://en.wikipedia.org/wiki/Inner_product_space) of 퐮 and 퐯. This is also known as the 2- norm. More generally, for 푝 ≥ 1, the 푝-norm (https://en.wikipedia.org/wiki/Lp_space#The_p- norm_in_countably_infinite_dimensions_and_ℓ_p_spaces) of 퐱 is given by 푑 1/푝 ‖퐱‖ = |푥 |푝 . 푝 (∑ 푖 ) 푖=1 Here (https://commons.wikimedia.org/wiki/File:Lp_space_animation.gif#/media/File:Lp_space_animation.gif) is a nice visualization of the unit ball, that is, the set {퐱 : ‖푥‖푝 ≤ 1}, under varying 푝. There exist many more norms. Formally: 푑 푑 Definition (Norm): A norm is a function ℓ from ℝ to ℝ+ that satisfies for all 푎 ∈ ℝ, 퐮, 퐯 ∈ ℝ (Homogeneity): ℓ(푎퐮) = |푎|ℓ(퐮) (Triangle inequality): ℓ(퐮 + 퐯) ≤ ℓ(퐮) + ℓ(퐯) (Point-separating): ℓ(푢) = 0 implies 퐮 = 0. ⊲ The triangle inequality for the 2-norm follows (https://en.wikipedia.org/wiki/Cauchy– Schwarz_inequality#Analysis) from the Cauchy–Schwarz inequality (https://en.wikipedia.org/wiki/Cauchy– Schwarz_inequality).
    [Show full text]
  • Continuous Probability Distributions Continuous Probability Distributions – a Guide for Teachers (Years 11–12)
    1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Probability and statistics: Module 21 Continuous probability distributions Continuous probability distributions – A guide for teachers (Years 11–12) Professor Ian Gordon, University of Melbourne Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Continuous random variables: basic ideas ....................... 5 Cumulative distribution functions ............................ 6 Probability density functions ..............................
    [Show full text]
  • 1 Probability Measure and Random Variables
    1 Probability measure and random variables 1.1 Probability spaces and measures We will use the term experiment in a very general way to refer to some process that produces a random outcome. Definition 1. The set of possible outcomes is called the sample space. We will typically denote an individual outcome by ω and the sample space by Ω. Set notation: A B, A is a subset of B, means that every element of A is also in B. The union⊂ A B of A and B is the of all elements that are in A or B, including those that∪ are in both. The intersection A B of A and B is the set of all elements that are in both of A and B. ∩ n j=1Aj is the set of elements that are in at least one of the Aj. ∪n j=1Aj is the set of elements that are in all of the Aj. ∩∞ ∞ j=1Aj, j=1Aj are ... Two∩ sets A∪ and B are disjoint if A B = . denotes the empty set, the set with no elements. ∩ ∅ ∅ Complements: The complement of an event A, denoted Ac, is the set of outcomes (in Ω) which are not in A. Note that the book writes it as Ω A. De Morgan’s laws: \ (A B)c = Ac Bc ∪ ∩ (A B)c = Ac Bc ∩ ∪ c c ( Aj) = Aj j j [ \ c c ( Aj) = Aj j j \ [ (1) Definition 2. Let Ω be a sample space. A collection of subsets of Ω is a σ-field if F 1.
    [Show full text]