miniModicum - Chapter 2 A modicum of measure theory SECTION...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 2 A modicum of measure theory SECTION 1 defines measures and sigma-fields. SECTION 2 defines measurable functions. SECTION 3 defines the integral with respect to a measure as a linear functional on a cone of measurable functions. The definition sidesteps the details of the construction of integrals from measures. SECTION *4 constructs integrals of nonnegative measurable functions with respect to a countably additive measure. SECTION 5 establishes the Dominated Convergence theorem, the Swiss Army knife of measure theoretic probability. SECTION 6 collects together a number of simple facts related to sets of measure zero. SECTION *7 presents a few facts about spaces of functions with integrable p th powers, with emphasis on the case p=2, which defines a Hilbert space. SECTION 8 defines uniform integrability, a condition slightly weaker than domination. Convergence in L 1 is characterized as convergence in probability plus uniform integrability. SECTION 9 defines the image measure, which includes the concept of the distribution of a random variable as a special case. SECTION 10 explains how generating class arguments, for classes of sets, make measure theory easy. SECTION *11 extends generating class arguments to classes of functions. 1. Measures and sigma-fields As promised in Chapter 1, we begin with measures as set functions, then work quickly towards the interpretation of integrals as linear functionals. Once we are past the purely set-theoretic preliminaries, I will start using the de Finetti notation (Section 1.4 ) in earnest, writing the same symbol for a set and its indicator function. Our starting point is a measure space : a triple ( X , A ,µ) , with X a set, A a class of subsets of X , and µ a function that attaches a nonnegative number (possibly +∞ ) to each set in A . The class A and the set function µ are required to have properties that facilitate calculations involving limits along sequences.
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
18 Chapter 2: A modicum of measure theory < 1 > Definition. Call a class A a sigma-field of subsets of X if: (i) the empty set and the whole space X both belong to A ; (ii) if A belongs to A then so does its complement A c ; (iii) if A 1 , A 2 ,... is a countable collection of sets in A then both the union i A i and the intersection i A i are also in A . Some of the requirements are redundant as stated. For example, once we have ∅ ∈ A then (ii) implies X A . When we come to establish properties about sigma-fields it will be convenient to have the list of defining properties pared down to a minimum, to reduce the amount of mechanical checking. The theorems will be as sparing as possible in the amount the work they require for establishing the sigma-field properties, but for now redundancy does not hurt. The collection A need not contain every subset of X , a fact forced upon us in general if we want µ to have the properties of a countably additive measure.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern