miniModicum

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

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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.

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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.
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