DOCSLIB.ORG

Math212a1413 the Lebesgue Integral

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math212a1413 the Lebesgue Integral Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. Math212a1413 The Lebesgue integral. Shlomo Sternberg October 28, 2014 Shlomo Sternberg Math212a1413 The Lebesgue integral. Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. Simple functions. In what follows, (X ; F; m) is a space with a σ-field of sets, and m a measure on F. The purpose of today's lecture is to develop the theory of the Lebesgue integral for functions defined on X . The theory starts with simple functions, that is functions which take on only finitely many non-zero values, say fa1;:::; ang and where −1 Ai := f (ai ) 2 F: In other words, we start with functions of the form n X φ(x) = ai 1Ai Ai 2 F: (1) i=1 Shlomo Sternberg Math212a1413 The Lebesgue integral. Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. The integral of simple functions. n X φ(x) = ai 1Ai Ai 2 F: i=1 Then, for any E 2 F we would like to define the integral of a simple function φ over E as n Z X φdm = ai m(Ai \ E) (2) E i=1 and extend this definition by some sort of limiting process to a broader class of functions. Shlomo Sternberg Math212a1413 The Lebesgue integral. Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. The range of the functions. I haven't specified what the range of the functions should be. Even to get started, we have to allow our functions to take values in a vector space over R, in order that the expression on the right of (2) make sense. I may eventually allow f to take values in a Banach space. However the theory is a bit simpler for real valued functions, where the linear order of the reals makes some arguments easier. Of course it would then be no problem to pass to any finite dimensional space over the reals. But we will on occasion need integrals in infinite dimensional Banach spaces, and that will require a little reworking of the theory. Shlomo Sternberg Math212a1413 The Lebesgue integral. Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. 1 Real valued measurable functions. 2 The integral of a non-negative function. 3 Fatou's lemma. 4 The monotone convergence theorem. 5 The space L1(X ; R). 6 The dominated convergence theorem. 7 Riemann integrability. 8 The Beppo-Levi theorem. 9 L1 is complete. 10 Dense subsets of L1(R; R). 11 The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. 12 Fubini's theorem. 13 The Borel transform. Shlomo Sternberg Math212a1413 The Lebesgue integral. Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. Measurable maps. If (X ; F) and (Y ; G) are spaces with σ-fields, then f : X ! Y is called measurable if f −1(E) 2 F 8 E 2 G: (3) Notice that the collection of subsets of Y for which (3) holds is a σ-field, and hence if it holds for some collection C, it holds for the σ-field generated by C. Shlomo Sternberg Math212a1413 The Lebesgue integral. Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. Measurable real valued functions. For the next few sections we will take Y = R and G = B, the Borel field. Since the collection of open intervals on the line generate the Borel field, a real valued function f : X ! R is measurable if and only if f −1(I ) 2 F for all open intervals I : Equally well, it is enough to check this for intervals of the form (−∞; a) for all real numbers a. Shlomo Sternberg Math212a1413 The Lebesgue integral. Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. Proposition 2 If F : R ! R is a continuous function and f ; g are two measurable real valued functions on X , then F (f ; g) is measurable. Proof. The set F −1(−∞; a) is an open subset of the plane, and hence can be written as the countable union of products of open intervals I × J. So if we set h = F (f ; g) then h−1((−∞; a)) is the countable union of the sets f −1(I ) \ g −1(J) and hence belongs to F. From this elementary proposition we conclude that if f and g are measurable real valued functions then: Shlomo Sternberg Math212a1413 The Lebesgue integral. Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. Sums, products, etc. f + g is measurable (since (x; y) 7! x + y is continuous), fg is measurable (since (x; y) 7! xy is continuous), hence f 1A is measurable for any A 2 F hence f + is measurable since f −1([0; 1]) 2 F and similarly for f − so jf j is measurable and so is jf − gj. Hence f ^ g and f _ g are measurable and so on. Shlomo Sternberg Math212a1413 The Lebesgue integral. Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. We are going to allow for the possibility that an integral might be infinite. We adopt the convention that 0 · 1 = 0: Recall that a non-negative function φ is simple if φ takes on a finite number of distinct non-negative values, a1;:::; an, and that each of the sets −1 Ai = φ (ai ) is measurable. These sets partition X : X = A1 [···[ An: Of course since the values are distinct, Ai \ Aj = ; for i 6= j: Shlomo Sternberg Math212a1413 The Lebesgue integral. Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. With this definition, a simple function can be written as in (1): n X φ(x) = ai 1Ai Ai 2 F (1) i=1 and this expression is unique. Shlomo Sternberg Math212a1413 The Lebesgue integral. Real valued measurable functions. The integral of a non-negative function. Fatou's lemma. The monotone convergence theorem. The space L1(X ; R). The dominated convergence theorem. Riemann integrability. The Beppo-Levi theorem. L1 is complete. Dense subsets of L1(R; R). The Riemann-Lebesgue Lemma and the Cantor-Lebesgue theorem. Fubini's theorem. The Borel transform. The integral of non-negative simple functions. A non-negative simple function can be written as in (1): n X φ(x) = ai 1Ai Ai 2 F (1) i=1 with ai ≥ 0 and this expression is unique. So we may take n Z X φdm = ai m(Ai \ E) (2) E i=1 as the definition of the integral of a non-negative simple function. Shlomo Sternberg Math212a1413 The Lebesgue integral.
Load more

Recommended publications
  • Integration 1 Measurable Functions
    Integration References: Bass (Real Analysis for Graduate Students), Folland (Real Analysis), Athreya and Lahiri (Measure Theory and Probability Theory). 1 Measurable Functions Let (Ω1; F1) and (Ω2; F2) be measurable spaces. Definition 1 A function T :Ω1 ! Ω2 is (F1; F2)-measurable if for every −1 E 2 F2, T (E) 2 F1. Terminology: If (Ω; F) is a measurable space and f is a real-valued func- tion on Ω, it's called F-measurable or simply measurable, if it is (F; B(<))- measurable. A function f : < ! < is called Borel measurable if the σ-algebra used on the domain and codomain is B(<). If the σ-algebra on the domain is Lebesgue, f is called Lebesgue measurable. Example 1 Measurability of a function is related to the σ-algebras that are chosen in the domain and codomain. Let Ω = f0; 1g. If the σ-algebra is P(Ω), every real valued function is measurable. Indeed, let f :Ω ! <, and E 2 B(<). It is clear that f −1(E) 2 P(Ω) (this includes the case where f −1(E) = ;). However, if F = f;; Ωg is the σ-algebra, only the constant functions are measurable. Indeed, if f(x) = a; 8x 2 Ω, then for any Borel set E containing a, f −1(E) = Ω 2 F. But if f is a function s.t. f(0) 6= f(1), then, any Borel set E containing f(0) but not f(1) will satisfy f −1(E) = f0g 2= F. 1 It is hard to check for measurability of a function using the definition, because it requires checking the preimages of all sets in F2.
    [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]
  • Measurable Functions and Simple Functions
    Measure theory class notes - 8 September 2010, class 9 1 Measurable functions and simple functions The class of all real measurable functions on (Ω, A ) is too vast to study directly. We identify ways to study them via simpler functions or collections of functions. Recall that L is the set of all measurable functions from (Ω, A ) to R and E⊆L are all the simple functions. ∞ Theorem. Suppose f ∈ L is bounded. Then there exists an increasing sequence {fn}n=1 in E whose uniform limit is f. Proof. First assume that f takes values in [0, 1). We divide [0, 1) into 2n intervals and use this to construct fn: n− 2 1 k − k k+1 fn = 1f 1 n , n 2n ([ 2 2 )) Xk=0 k k+1 k Whenever f takes a value in 2n , 2n , fn takes the value 2n . We have • For all n, fn ≤ f. 1 • For all n, x, |fn(x) − f(x)|≤ 2n . This is clear from the construction. • fn ∈E, since fn is a finite linear combination of indicator functions of sets in A . k 2k 2k+1 • fn ≤ fn+1: For any x, if fn(x)= 2n , then fn+1(x) ∈ 2n+1 , 2n+1 . ∞ So {fn}n=1 is an increasing sequence in E converging uniformly to f. Now for a general f, if the image of f lies in [a,b), then let f − a1 g = Ω b − a (note that a1Ω is the constant function a) ∞ Im g ⊆ [0, 1), so by the above we have {gn}n=1 from E increasing uniformly to g.
    [Show full text]
  • Lecture 3 the Lebesgue Integral
    Lecture 3:The Lebesgue Integral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Instructor: Gordan Zitkovic Lecture 3 The Lebesgue Integral The construction of the integral Unless expressly specified otherwise, we pick and fix a measure space (S, S, m) and assume that all functions under consideration are defined there. Definition 3.1 (Simple functions). A function f 2 L0(S, S, m) is said to be simple if it takes only a finite number of values. The collection of all simple functions is denoted by LSimp,0 (more precisely by LSimp,0(S, S, m)) and the family of non-negative simple Simp,0 functions by L+ . Clearly, a simple function f : S ! R admits a (not necessarily unique) representation n f = ∑ ak1Ak , (3.1) k=1 for a1,..., an 2 R and A1,..., An 2 S. Such a representation is called the simple-function representation of f . When the sets Ak, k = 1, . , n are intervals in R, the graph of the simple function f looks like a collection of steps (of heights a1,..., an). For that reason, the simple functions are sometimes referred to as step functions. The Lebesgue integral is very easy to define for non-negative simple functions and this definition allows for further generalizations1: 1 In fact, the progression of events you will see in this section is typical for measure theory: you start with indica- Definition 3.2 (Lebesgue integration for simple functions). For f 2 tor functions, move on to non-negative Simp,0 R simple functions, then to general non- L+ we define the (Lebesgue) integral f dm of f with respect to m negative measurable functions, and fi- by nally to (not-necessarily-non-negative) Z n f dm = a m(A ) 2 [0, ¥], measurable functions.
    [Show full text]
  • Measure, Integrals, and Transformations: Lebesgue Integration and the Ergodic Theorem
    Measure, Integrals, and Transformations: Lebesgue Integration and the Ergodic Theorem Maxim O. Lavrentovich November 15, 2007 Contents 0 Notation 1 1 Introduction 2 1.1 Sizes, Sets, and Things . 2 1.2 The Extended Real Numbers . 3 2 Measure Theory 5 2.1 Preliminaries . 5 2.2 Examples . 8 2.3 Measurable Functions . 12 3 Integration 15 3.1 General Principles . 15 3.2 Simple Functions . 20 3.3 The Lebesgue Integral . 28 3.4 Convergence Theorems . 33 3.5 Convergence . 40 4 Probability 41 4.1 Kolmogorov's Probability . 41 4.2 Random Variables . 44 5 The Ergodic Theorem 47 5.1 Transformations . 47 5.2 Birkho®'s Ergodic Theorem . 52 5.3 Conclusions . 60 6 Bibliography 63 0 Notation We will be using the following notation throughout the discussion. This list is included here as a reference. Also, some of the concepts and symbols will be de¯ned in subsequent sections. However, due to the number of di®erent symbols we will use, we have compiled the more archaic ones here. lR; lN : the real and natural numbers lR¹ : the extended natural numbers, i.e. the interval [¡1; 1] ¹ lR+ : the nonnegative (extended) real numbers, i.e. the interval [0; 1] ­ : a sample space, a set of possible outcomes, or any arbitrary set F : an arbitrary σ-algebra on subsets of a set ­ σ hAi : the σ-algebra generated by a subset A ⊆ ­. T : a function between the set ­ and ­ itself Á, Ã : simple functions on the set ­ ¹ : a measure on a space ­ with an associated σ-algebra F (Xn) : a sequence of objects Xi, i.
    [Show full text]
  • Lecture 4 Chapter 1.4 Simple Functions
    Math 103: Measure Theory and Complex Analysis Fall 2018 09/19/18 Lecture 4 Chapter 1.4 Simple functions Aim: A function is Riemann integrable if it can be approximated with step functions. A func- tion is Lebesque integrable if it can be approximated with measurable simple functions.A measurable simple function is similar to a step function, just that the supporting sets are ele- ments of a σ algebra. Picture Denition 1 (Simple functions) A function s : X ! C is called a simple function if it has nite range. We say that s is a non-negative simple function (nnsf) if s(X) ⊂ [0; +1). Note 2 If s(X) 6= f0g, then s(X) = fa1; a2; a3; : : : ; ang and let Ai = fx 2 X j s(x) = aig. Then s is measurable if and only if Ai 2 M for all i 2 f1; 2; : : : ; ng. In this case we have that n X 1 (1) s = ai · Ai : i=1 We can furthermore assume that the (Ai)i are mutually disjoint. This representation as a linear combination of characteristic functions is unique, if the (ai)i are distinct and non-zero. In this case we call it the standard representation of s. Theorem 3 (Approximation by simple functions) For any function f :(X; M) ! [0; +1], there are nnsfs (sn)n2N on X, such that a) 0 ≤ s1 ≤ s2 ≤ ::: ≤ f. b) For all x 2 X, we have limn!1 sn(x) = f(x). Furthermore, if f is measurable, then the (sn)n can be chosen measurable as well.
    [Show full text]
  • Representations and Positive Spectral Measures
    Master's Thesis Positive C(K)-representations and positive spectral measures On their one-to-one correspondence for reflexive Banach lattices Author: Supervisor: Frejanne Ruoff dr. M.F.E. de Jeu Defended on January 30, 2014 Leiden University Faculty of Science Mathematics Specialisation: Applied Mathematics Mathematical Institute, Leiden University Abstract Is there a natural one-to-one correspondence between positive representations of spaces of continuous functions and positive spectral measures on Banach lattices? That is the question that we have investigated in this thesis. Representations and spectral measures are key notions throughout all sections. After having given precise definitions of a positive spectral measure, a unital positive spectral measure, a positive representation, and a unital positive representation, we prove that there is a one-to-one relationship between the positive spectral measures and positive representations, and the unital positive spectral measures and unital positive representations, respectively. For example, between a unital positive representation ρ : C(K) !Lr(X) and a unital positive spectral measure E :Ω !Lr(X), where K is a compact Hausdorff space, Ω ⊆ P(K) the Borel σ-algebra, X a reflexive Banach lattice, and Lr(X) the space of all regular operators on X. The map ρ : C(K) !Lr(C(K)), where K is a compact Hausdorff space, defined as pointwise multiplication on C(K), is a unital positive representation. The subspace ρ[f] := ρ(C(K))f ⊆ C(K), for an f 2 C(K), has two partial orderings. One is inherited from C(K), the other is newly defined. Using the defining property of elements of ρ[f], we prove that they are equal.
    [Show full text]
  • 1. Σ-Algebras Definition 1. Let X Be Any Set and Let F Be a Collection Of
    1. σ-algebras Definition 1. Let X be any set and let F be a collection of subsets of X. We say that F is a σ-algebra (on X), if it satisfies the following. (1) X 2 F. (2) If A 2 F, then Ac 2 F. 1 (3) If A1;A2; · · · 2 F, a countable collection, then [n=1An 2 F. In the above situation, the elements of F are called measurable sets and X (or more precisely (X; F)) is called a measurable space. Example 1. For any set X, P(X), the set of all subsets of X is a σ-algebra and so is F = f;;Xg. Theorem 1. Let X be any set and let G be any collection of subsets of X. Then there exists a σ-algebra, containing G and smallest with respect to inclusion. Proof. Let S be the collection of all σ-algebras containing G. This set is non-empty, since P(X) 2 S. Then F = \A2SA can easily be checked to have all the properties asserted in the theorem. Remark 1. In the above situation, F is called the σ-algebra generated by G. Definition 2. Let X be a topological space (for example, a metric space) and let B be the σ-algebra generated by the set of all open sets of X. The elements of B are called Borel sets and (X; B), a Borel measurable space. Definition 3. Let (X; F) be a measurable space and let Y be any topo- logical space and let f : X ! Y be any function.
    [Show full text]
  • Real Analysis a Comprehensive Course in Analysis, Part 1
    Real Analysis A Comprehensive Course in Analysis, Part 1 Barry Simon Real Analysis A Comprehensive Course in Analysis, Part 1 http://dx.doi.org/10.1090/simon/001 Real Analysis A Comprehensive Course in Analysis, Part 1 Barry Simon Providence, Rhode Island 2010 Mathematics Subject Classification. Primary 26-01, 28-01, 42-01, 46-01; Secondary 33-01, 35-01, 41-01, 52-01, 54-01, 60-01. For additional information and updates on this book, visit www.ams.org/bookpages/simon Library of Congress Cataloging-in-Publication Data Simon, Barry, 1946– Real analysis / Barry Simon. pages cm. — (A comprehensive course in analysis ; part 1) Includes bibliographical references and indexes. ISBN 978-1-4704-1099-5 (alk. paper) 1. Mathematical analysis—Textbooks. I. Title. QA300.S53 2015 515.8—dc23 2014047381 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 select pages 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. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright.
    [Show full text]
  • Probability Theory
    Probability Theory "A random variable is neither random nor variable." Gian-Carlo Rota, M.I.T.. Florian Herzog 2013 Probability space Probability space A probability space W is a unique triple W = fΩ; F;P g: • Ω is its sample space •F its σ-algebra of events • P its probability measure Remarks: (1) The sample space Ω is the set of all possible samples or elementary events !: Ω = f! j ! 2 Ωg. (2)The σ-algebra F is the set of all of the considered events A, i.e., subsets of Ω: F = fA j A ⊆ Ω;A 2 Fg. (3) The probability measure P assigns a probability P (A) to every event A 2 F: P : F! [0; 1]. Stochastic Systems, 2013 2 Sample space The sample space Ω is sometimes called the universe of all samples or possible outcomes !. Example 1. Sample space • Toss of a coin (with head and tail): Ω = fH;T g. • Two tosses of a coin: Ω = fHH;HT;TH;TT g. • A cubic die: Ω = f!1;!2;!3;!4;!5;!6g. • The positive integers: Ω = f1; 2; 3;::: g. • The reals: Ω = f! j ! 2 Rg. Note that the !s are a mathematical construct and have per se no real or scientific meaning. The !s in the die example refer to the numbers of dots observed when the die is thrown. Stochastic Systems, 2013 3 Event An event A is a subset of Ω. If the outcome ! of the experiment is in the subset A, then the event A is said to have occurred.
    [Show full text]
  • Real Analysis
    Real Analysis Part I: MEASURE THEORY 1. Algebras of sets and σ-algebras For a subset A ⊂ X, the complement of A in X is written X − A. If the ambient space X is understood, in these notes we will sometimes write Ac for X − A. In the literature, the notation A′ is also used sometimes, and the textbook uses A˜ for the complement of A. The set of subsets of a set X is called the power set of X, written 2X . Definition. A collection F of subsets of a set X is called an algebra of sets in X if 1) ∅,X ∈F 2) A ∈F⇒ Ac ∈F 3) A,B ∈F⇒ A ∪ B ∈F In mathematics, an “algebra” is often used to refer to an object which is simulta- neously a ring and a vector space over some field, and thus has operations of addition, multiplication, and scalar multiplication satisfying certain compatibility conditions. How- ever the use of the word “algebra” within the phrase “algebra of sets” is not related to the preceding use of the word, but instead comes from its appearance within the phrase “Boolean algebra”. A Boolean algebra is not an algebra in the preceding sense of the word, but instead an object with operations AND, OR, and NOT. Thus, as with the phrase “Boolean algebra”, it is best to treat the phrase “algebra of sets” as a single unit rather than as a sequence of words. For reference, the definition of a Boolean algebra follows. Definition. A Boolean algebra consists of a set Y together with binary operations ∨, ∧, and a unariy operation ∼ such that 1) x ∨ y = y ∨ x; x ∧ y = y ∧ x for all x,y ∈ Y 2) (x ∨ y) ∨ z = x ∨ (y ∨ z); (x ∧ y) ∧ z = x ∧ (y ∧ z) forall x,y,z ∈ Y 3) x ∨ x = x; x ∧ x = x for all x ∈ Y 4) (x ∨ y) ∧ x = x; (x ∧ y) ∨ x = x for all x ∈ Y 5) x ∧ (y ∨ z)=(x ∧ y) ∨ (x ∧ z) forall x,y,z ∈ Y 6) ∃ an element 0 ∈ Y such that 0 ∧ x = 0 for all x ∈ Y 7) ∃ an element 1 ∈ Y such that 1 ∨ x = 1 for all x ∈ Y 8) x ∧ (∼ x)=0; x ∨ (∼ x)=1 forall x ∈ Y Other properties, such as the other distributive law x ∨ (y ∧ z)=(x ∨ y) ∧ (x ∨ z), can be derived from the ones listed.
    [Show full text]