DOCSLIB.ORG

Math 3345-Real Analysis — Lecture 19 10/26/05 1. Counting Measures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 3345-Real Analysis — Lecture 19 10/26/05 1. Counting Measures Math 3345-Real Analysis — Lecture 19 10/26/05 1. Counting Measures and Cardinality Derivatives and integrals depend heavily on the notion of length. The essence of the relevant concepts depends on the concept of measure. Let’s start with a simple measure. Question. How big is the set { 1, 3, 4, 7 }? Well, this set has four elements. What else is there to say? We can measure the size of finite sets by counting how many elements there are in the set. This is known as a counting measure, and if you can measure sets, then it’s possible to talk about derivatives and integrals. Let’s see how that works. Consider the function f : Z → R defined by f(x)=x2. Note that this function is only defined on Z. What’s the derivative of this function? For functions defined on R, the derivative is a rate of change. For example, if f 0(2) = 5, then the slope of the tangent line is m =5atx = 2, and y is increasing by 5 as x increases by 1. In this case, what should f 0(2) be? Well, f(2) = 4 and f(3) = 9, so as x changes from 2 → 3, y changes from 4 → 9. In other words, y compared to x is changing at a rate of 5 to 1. f 0(2) must be 5. Let’s define this in general. The derivative of a function f : Z → R at x is f(x +1)− f(x) (1) f 0(x)= . 1 Here we get f 0(0) = 1 f 0(1) = 3 (2) f 0(2) = 5 f 0(3) = 7 It looks like (3) f 0(x)=2x +1. This may look odd, but we’ve got a different kind of derivative. In regular differentiation and integration, we have the fundamental theorem of calculus. It looks like b (4) Z f 0(x) dx = f(b) − f(a). a Do we have something similar with functions defined on Z? Let’s take a look. How should 2 (5) Z f 0(x) dx 0 be calculated? With the Riemann integeral, we multiplied function values times little lengths (like ∆x)in the domain. Then we added everything up. We don’t have little lengths in our domain, but we have the counting measure, which is the same sort of thing. Each point in the domain counts as 1. Therefore, 2 (6) Z f 0(x) dx = f 0(0) · 1+f 0(1) · 1+f 0(2) · 1=1+3+5=9. 0 In other words, with a counting measure, we just add up the function values. Note that the 9 looks familiar. 2 (7) Z f 0(x) dx =9=f(3) − f(0). 0 1 2 Does this make sense? Of course. The function f increases by 1 as x goes from 0 to 1. It increases by 3 as x goes from 1 to 2. It increases by 5 as x goes from 2 to 3. There is a total increase of 1 + 3 + 5 = 9. We also know that f increases from 0 to 9 as x goes from 0 to 3. 7 0 1. Compute R2 f (x) dx. 2. Compute f(8) − f(2). Is the real fundamental theorem of calculus this simple? Yes. Well, actually no. But the basic idea holds. 2. Cardinality Counting the number of elements in a set measures a set’s cardinality. One way of describing this is as follows. Consider the set { 1, 3, 4, 7 }. We count the elements one, two, three, four. In other words, we form a one-to-one correspondence 1 → 1 2 → 3 (8) 3 → 4 4 → 7 A finite set will have the same cardinality as one of the following sets: {}= ∅ { 1 } { 1, 2 } { } (9) 1, 2, 3 { 1, 2, 3, 4 } { 1, 2, 3, 4, 5 } . This corresponds to saying how many elements are in the set. An infinite set could have the same cardinality as the natural numbers N = { 1, 2, 3, 4, 5,...}. In this case, we’ll say the set is countably infinite, or just countable. The integers are countable. An indication of this is the correspondence below. 1 → 0 2 → 1 3 →−1 4 → 2 (10) 5 →−2 6 → 3 . Essentially, this indicates how all the integers will fit into a sequence. 3 The rationals are also countable. The correspondence below puts all the positive rationals into a sequence. 1 1 → 1 1 2 → 2 2 3 → 1 1 4 → (11) 3 2 5 → 2 3 6 → 1 1 7 → 4 . Note that this takes all the fractions whose denominator and numerator add up to two, then all the fractions whose denominator and numerator add up to three, then four, then five, etc. Can you see how every positive fraction must be in this list? Actually, there are lot’s of repetitions, so each positive fraction is in the list at least once. It should be intuitively apparent that we can skip over the repetitions to get a one-to-one correspondence. We could also play the same trick used with the integers to get the negative rationals into the list. The reals are not countable. The reals between 0 and 1 are not countable. To see this, suppose that we could put all the reals between 0 and 1 into a list. Each of these numbers has a decimal expansion, and it might look something like this. We’ll also make sure that we don’t use the repeating 9’s representation for any number. 1 → 0.458796542145544 ... 2 → 0.578945457856215 ... 3 → 0.145872525463245 ... 4 → 0.333333333333333 ... (12) 5 → 0.668877445587445 ... 6 → 0.000014587544587 ... 7 → 0.999948754745555 ... From here, it’s easy to construct the decimal expansion of a number not on the list. Since the first digit of the first number is 4, I’ll make sure that the first digit of my number isn’t a 4. How about 7? The second digit of the second number is 7, so I’ll pick 2 for the second digit of my number. The third digit of the third number is 5, so I’ll pick 8 for the third digit of my number. Now, I could get messed up, if I pick all 0’s or all 9’s, so I’ll never pick a 0 or a 9. So I’ve got (13) x =0.728 ... and x is not on the list. I can do this for any list, so it must be impossible to get all the real numbers into a sequence. The reals are not countable. We’ll say that the reals are uncountable. 4 3. Summary of cardinality We can specifically count the number of elements in a finite set. There are countably infinite sets, like Q and Z, and there are uncountable sets, like R and I. It is relatively easy to show that there are sets, such as the set of all real-valued functions, that have cardinalities larger than R’s. In fact, given a set, it’s “easy” to construct a set with a larger cardinality. So, we’ve got all the finite cardinalities, the countably infinite cardinality, and an infinite number of uncountable cardinalities. We could ask the question, is there a cardinality between countably infinite and the cardinality of R? This question is known to be unanswerable. We can adopt the following Continuum Hypothesis, which we generally do, or we can assume that it’s false. Either way, there is no contradiction with the normal “facts” of mathematics. The Continuum Hypothesis. The cardinality of the real numbers is the smallest uncountable cardinality. Homework 19 starts on the next page. 5 4. Homework 19 1. The correspondence between N and Q+ that I have above has repetitions. If you were to remove the repetitions, what would be the 20th fraction in the list? 2. Your list from problem 1 could generate a list for the negative rationals. We could then get all of Q into a list by starting with 0, taking the first thing in the “positive” list, then the first thing in the “negative” list, then the second thing in the “positive” list, etc. What’s the fifteenth thing on this list? We can’t talk about uncountable cardinalities with lists. To see a one-to-one correspondence with the reals, consider the graph of the tangent function. One brach of this function sets up a one-to-one correspondence − π π between 2 , 2 and R. Once you buy that pile of canned goods, you have to accept that any open interval has the same cardinality as the real numbers. To work with cardinalities, it’s convenient to nail down some intuitive axioms. For a set A, we will denote the cardinality of A by | A |. 3. Would you be willing to accept Axiom I? If A ⊂ B, then | A |≤|B |. 4. Would you be willing to accept Axiom II? If A ⊂ B and A =6 B (i.e., A is a proper subset of B), then the cardinality of A could be the same as B’s. 5. Would you be willing to accept Axiom III? Given two sets A and B, then exactly one of the following is true: | A | < | B |, | A | = | B |,or| A | > | B |. 6. What axiom says that | I |≤|R |? If I were countable, then we would have R as the union of two countable sets. If this were true, then both Q and I could be put into sequences, and then alternating terms from these two sequences would put all of R into a sequence.
Load more

Recommended publications
  • Measure-Theoretic Probability I
    Measure-Theoretic Probability I Steven P.Lalley Winter 2017 1 1 Measure Theory 1.1 Why Measure Theory? There are two different views – not necessarily exclusive – on what “probability” means: the subjectivist view and the frequentist view. To the subjectivist, probability is a system of laws that should govern a rational person’s behavior in situations where a bet must be placed (not necessarily just in a casino, but in situations where a decision must be made about how to proceed when only imperfect information about the outcome of the decision is available, for instance, should I allow Dr. Scissorhands to replace my arthritic knee by a plastic joint?). To the frequentist, the laws of probability describe the long- run relative frequencies of different events in “experiments” that can be repeated under roughly identical conditions, for instance, rolling a pair of dice. For the frequentist inter- pretation, it is imperative that probability spaces be large enough to allow a description of an experiment, like dice-rolling, that is repeated infinitely many times, and that the mathematical laws should permit easy handling of limits, so that one can make sense of things like “the probability that the long-run fraction of dice rolls where the two dice sum to 7 is 1/6”. But even for the subjectivist, the laws of probability should allow for description of situations where there might be a continuum of possible outcomes, or pos- sible actions to be taken. Once one is reconciled to the need for such flexibility, it soon becomes apparent that measure theory (the theory of countably additive, as opposed to merely finitely additive measures) is the only way to go.
    [Show full text]
  • Measure Theory and Probability
    Measure theory and probability Alexander Grigoryan University of Bielefeld Lecture Notes, October 2007 - February 2008 Contents 1 Construction of measures 3 1.1Introductionandexamples........................... 3 1.2 σ-additive measures ............................... 5 1.3 An example of using probability theory . .................. 7 1.4Extensionofmeasurefromsemi-ringtoaring................ 8 1.5 Extension of measure to a σ-algebra...................... 11 1.5.1 σ-rings and σ-algebras......................... 11 1.5.2 Outermeasure............................. 13 1.5.3 Symmetric difference.......................... 14 1.5.4 Measurable sets . ............................ 16 1.6 σ-finitemeasures................................ 20 1.7Nullsets..................................... 23 1.8 Lebesgue measure in Rn ............................ 25 1.8.1 Productmeasure............................ 25 1.8.2 Construction of measure in Rn. .................... 26 1.9 Probability spaces ................................ 28 1.10 Independence . ................................. 29 2 Integration 38 2.1 Measurable functions.............................. 38 2.2Sequencesofmeasurablefunctions....................... 42 2.3 The Lebesgue integral for finitemeasures................... 47 2.3.1 Simplefunctions............................ 47 2.3.2 Positivemeasurablefunctions..................... 49 2.3.3 Integrablefunctions........................... 52 2.4Integrationoversubsets............................ 56 2.5 The Lebesgue integral for σ-finitemeasure.................
    [Show full text]
  • An Approximate Logic for Measures
    AN APPROXIMATE LOGIC FOR MEASURES ISAAC GOLDBRING AND HENRY TOWSNER Abstract. We present a logical framework for formalizing connections be- tween finitary combinatorics and measure theory or ergodic theory that have appeared in various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence as well as provide short proofs of the Szemer´edi Regularity Lemma and the hypergraph removal lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics. 1. Introduction Since the 1970’s, diagonalization arguments (and their generalization to ul- trafilters) have been used to connect results in finitary combinatorics with results in measure theory or ergodic theory [14]. Although it has long been known that this connection can be described with first-order logic, a recent striking paper by Hrushovski [22] demonstrated that modern developments in model theory can be used to give new substantive results in this area as well. Papers on various aspects of this interaction have been published from a va- riety of perspectives, with almost as wide a variety of terminology [30, 10, 9, 1, 32, 13, 4]. Our goal in this paper is to present an assortment of these techniques in a common framework. We hope this will make the entire area more accessible. We are typically concerned with the situation where we wish to prove a state- ment 1) about sufficiently large finite structures which 2) concerns the finite analogs of measure-theoretic notions such as density or integrals.
    [Show full text]
  • Dynamic Ham-Sandwich Cuts in the Plane∗
    Dynamic Ham-Sandwich Cuts in the Plane∗ Timothy G. Abbott† Michael A. Burr‡ Timothy M. Chan§ Erik D. Demaine† Martin L. Demaine† John Hugg¶ Daniel Kanek Stefan Langerman∗∗ Jelani Nelson† Eynat Rafalin†† Kathryn Seyboth¶ Vincent Yeung† Abstract We design efficient data structures for dynamically maintaining a ham-sandwich cut of two point sets in the plane subject to insertions and deletions of points in either set. A ham- sandwich cut is a line that simultaneously bisects the cardinality of both point sets. For general point sets, our first data structure supports each operation in O(n1/3+ε) amortized time and O(n4/3+ε) space. Our second data structure performs faster when each point set decomposes into a small number k of subsets in convex position: it supports insertions and deletions in O(log n) time and ham-sandwich queries in O(k log4 n) time. In addition, if each point set has convex peeling depth k, then we can maintain the decomposition automatically using O(k log n) time per insertion and deletion. Alternatively, we can view each convex point set as a convex polygon, and we show how to find a ham-sandwich cut that bisects the total areas or total perimeters of these polygons in O(k log4 n) time plus the O((kb) polylog(kb)) time required to approximate the root of a polynomial of degree O(k) up to b bits of precision. We also show how to maintain a partition of the plane by two lines into four regions each containing a quarter of the total point count, area, or perimeter in polylogarithmic time.
    [Show full text]
  • 21 TOPOLOGICAL METHODS in DISCRETE GEOMETRY Rade T
    21 TOPOLOGICAL METHODS IN DISCRETE GEOMETRY Rade T. Zivaljevi´cˇ INTRODUCTION A problem is solved or some other goal achieved by “topological methods” if in our arguments we appeal to the “form,” the “shape,” the “global” rather than “local” structure of the object or configuration space associated with the phenomenon we are interested in. This configuration space is typically a manifold or a simplicial complex. The global properties of the configuration space are usually expressed in terms of its homology and homotopy groups, which capture the idea of the higher (dis)connectivity of a geometric object and to some extent provide “an analysis properly geometric or linear that expresses location directly as algebra expresses magnitude.”1 Thesis: Any global effect that depends on the object as a whole and that cannot be localized is of homological nature, and should be amenable to topological methods. HOW IS TOPOLOGY APPLIED IN DISCRETE GEOMETRIC PROBLEMS? In this chapter we put some emphasis on the role of equivariant topological methods in solving combinatorial or discrete geometric problems that have proven to be of relevance for computational geometry and topological combinatorics and with some impact on computational mathematics in general. The versatile configuration space/test map scheme (CS/TM) was developed in numerous research papers over the years and formally codified in [Ziv98].ˇ Its essential features are summarized as follows: CS/TM-1: The problem is rephrased in topological terms. The problem should give us a clue how to define a “natural” configuration space X and how to rephrase the question in terms of zeros or coincidences of the associated test maps.
    [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]
  • Measure and Integration
    Chapter 5 Measure and integration In calculus you have learned how to calculate the size of different kinds of sets: the length of a curve, the area of a region or a surface, the volume or mass of a solid. In probability theory and statistics you have learned how to compute the size of other kinds of sets: the probability that certain events happen or do not happen. In this chapter we shall develop a general theory for the size of sets, a theory that covers all the examples above and many more. Just as the concept of a metric space gave us a general setting for discussing the notion of distance, the concept of a measure space will provide us with a general setting for discussing the notion of size. In calculus we use integration to calculate the size of sets. In this chap- ter we turn the situation around: We first develop a theory of size and then use it to define integrals of a new and more general kind. As we shall sometimes wish to compare the two theories, we shall refer to integration as taught in calculus as Riemann-integration in honor of the German math- ematician Bernhard Riemann (1826-1866) and the new theory developed here as Lebesgue integration in honor of the French mathematician Henri Lebesgue (1875-1941). Let us begin by taking a look at what we might wish for in a theory of size. Assume what we want to measure the size of subsets of a set X (if 2 you need something concrete to concentrate on, you may let X = R and 2 3 think of the area of subsets of R , or let X = R and think of the volume of 3 subsets of R ).
    [Show full text]
  • Measure Theory 1 Measurable Spaces
    Measure Theory 1 Measurable Spaces A measurable space is a set S, together with a nonempty collection, , of subsets of S, satisfying the following two conditions: S 1. For any A; B in the collection , the set1 A B is also in . S − S 2. For any A1; A2; , Ai . The elements of ·are· · 2calledS [measurable2 S sets. These two conditions are S summarized by saying that the measurable sets are closed under taking finite differences and countable unions. Think of S as the arena in which all the action (integrals, etc) will take place; and of the measurable sets are those that are \candidates for having a size". In some examples, all the measurable sets will be assigned a \size"; in others, only the smaller measurable sets will be (with the remaining mea- surable sets having, effectively “infinite size"). Several properties of measurable sets are immediate from the definition. 1. The empty set, , is measurable. [Since is nonempty, there exists some measurable set A.;So, A A = is measurable,S by condition 1 above.] − ; 2. For A and B any two measurable sets, A B, A B, and A B are all measurable. [The third is just condition 1\above.[For the second,− apply condition 2 to the sequence A; B; ; ; . For the first, note that ; ; · · · A B = A (A B): Use condition 1 twice.] It follows immediately, by rep\eated application− − of these facts, that the measurable sets are closed under taking any finite numbers of intersections, unions, and differences. 3. For A1; A2; measurable, their intersection, A , is also measurable.
    [Show full text]
  • Notes on Measure Theory and the Lebesgue Integral Maa5229, Spring 2015
    NOTES ON MEASURE THEORY AND THE LEBESGUE INTEGRAL MAA5229, SPRING 2015 1. σ-algebras Let X be a set, and let 2X denote the set of all subsets of X. Let Ec denote the complement of E in X, and for E; F ⊂ X, write E n F = E \ F c. Let E∆F denote the symmetric difference of E and F : E∆F := (E n F ) [ (F n E) = (E [ F ) n (E \ F ): Definition 1.1. Let X be a set. A Boolean algebra is a nonempty collection A ⊂ 2X which is closed under finite unions and complements. A σ-algebra is a Boolean algebra which is also closed under countable unions. If M ⊂ N ⊂ 2X are σ-algebras, then M is coarser than N ; likewise N is finer than M . / Remark 1.2. If Eα is any collection of sets in X, then !c [ c \ Eα = Eα: (1) α α Hence a Boolean algebra (resp. σ-algebra) is automatically closed under finite (resp. countable) intersections. It follows that a Boolean algebra (and a σ-algebra) on X always contains ? and X. (Proof: X = E [ Ec and ? = E \ Ec.) Definition 1.3. A measurable space is a pair (X; M ) where M ⊂ 2X is a σ-algebra. A function f : X ! Y from one measurable spaces (X; M ) to another (Y; N ) is called measurable if f −1(E) 2 M whenever E 2 N . / Definition 1.4. A topological space X = (X; τ) consists of a set X and a subset τ of 2X such that (i) ;;X 2 τ; (ii) τ is closed under finite intersections; (iii) τ is closed under arbitrary unions.
    [Show full text]
  • Lecture Notes 9: Measure and Probability Part A: Measure and Integration
    Lecture Notes 9: Measure and Probability Part A: Measure and Integration Peter Hammond, based on early notes by Andr´esCarvajal September, 2011; latest revision September 2020, typeset from Probab20A.tex; Recommended textbooks for further reading: H.L. Royden Real Analysis; R.M. Dudley Real Analysis and Probability University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Outline Measures and Integrals Measurable Spaces Measure Spaces Lebesgue Integration Integrating Simple Functions Integrating Integrable Functions The Lebesgue Integral as an Antiderivative Leibnitz's Formula Revisited Products of Measure Spaces Definition The Gaussian Integral Circular Symmetry and Shell Integration University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 2 of 57 Power Sets Fix an abstract set S 6= ;. In case it is finite, its cardinality, denoted by #S, is the number of distinct elements of S. The power set of S is the family P(S) := fT j T ⊆ Sg of all subsets of S. Exercise Show that the mapping S P(S) 3 T 7! f (T ) 2 f0; 1g := fhxs is2S j 8s 2 S : xs 2 f0; 1gg defined by ( 1 if s 2 T f (T )s = 1T (s) = 0 if s 62 T is a bijection. University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 3 of 57 Constructing a Bijection Proof. S Evidently the mapping T 7! f (T ) = h1T (s)is2S 2 f0; 1g defines a unique point of the Cartesian product f0; 1gS for every T ⊆ S | i.e., for every T 2 P(S). S Conversely, for each hxs is2S 2 f0; 1g , there is a unique set T (hxs is2S ) = fs 2 S j xs = 1g Now, given this hxs is2S , for each r 2 S, one has ( 1 if xr = 1 1fs2Sjxs =1g(r) = 0 if xr = 0 So f (T (hxs is2S )) = h1fs2Sjxs =1g(r)ir2S = hxr ir2S .
    [Show full text]
  • Chapter 17. General Measure Spaces: Their Properties and Construction
    17.1. Measures and Measurable Sets 1 Chapter 17. General Measure Spaces: Their Properties and Construction Note. In Chapter 2, we defined Lebesgue measure on the σ-algebra M, and used open sets (open intervals, specifically) in the definition. In this chapter, we give an abstract definition of a measure on a σ-algebra of some point set. The technique is analogous to the definition of a topology—the sets which are open are, by definition, the sets in the topology. Hence, the sets which are measurable are, by definition, the sets in the σ-algebra. We will still, however, have some analogies with outer measure and the Carath´eodory splitting condition. Section 17.1. Measures and Measurable Sets Note. In this section, we define a measure space and show parallels between this new setting and the results of Chapter 2. Definition. A measurable space is an ordered pair (X, M) consisting of a set X and a σ-algebra M of subsets of X. Set E ⊂ X is measurable if E ∈ M. A measure µ on a measurable space (X, M) is an extended real-valued nonnegative set function µ : M→ [0, ∞] for which µ(∅) = 0 and which is countably additive in ∞ the sense that for any countable disjoint collection {Ek}k=1 of measurable sets, ∞ ∞ µ · Ek = µ(Ek). k=1 ! k=1 [ X A measure space (X, M, µ) is a measurable space (X, M) together with a measure µ on M. 17.1. Measures and Measurable Sets 2 Note. We denote the power set of X, motivated by properties of cardinal numbers, as P(X) = 2X .
    [Show full text]
  • Lecture 24: Properties of Measures
    Lecture 24: Properties of Measures 1 Properties of Measures We will assume (X; F) to be the measurable space throughout this lecture, unless specified otherwise. Definition 1.1. Let (X; F) be a measurable space. A measure µ : F! [0; 1] is called 1.a probability measure, if µ(X) = 1, 2.a finite measure, if µ(X) < 1, 3.a σ-finite measure, if there exists a sequence fAn 2 F : n 2 Ng such that [n2NAn = X and µ(An) < 1 for all n 2 N, 4.a semi-finite measure, if for each E 2 F with µ(E) = 1, there exists F 2 F with F ⊆ E and µ(F ) < 1, 5. diffuse or atom-free, if µ(fxg) = 0, whenever x 2 X and fxg 2 F. Lemma 1.2. If µ is a finite measure, then µ(E) is finite for all E 2 F. Proof. Let E 2 F, then X n E 2 F. Further, set X can be expressed as disjoint union of E t (X n E). Then, by finite additivity of measures, we have µ(X) = µ(E) + µ(X n E). Hence, by non-negativity of measures, we have µ(E) < 1. Definition 1.3. Let (X; F; µ) be a measure space. If E = [j2NEj where µ(Ej) < 1 for all j 2 N, then the set E is called of σ-finite measure. Lemma 1.4. A probability measure is finite. A finite measure is σ-finite. Proposition 1.5. Every σ-finite measure is semi-finite.
    [Show full text]