Product

Product - Chapter 4 Product spaces and independence 1 <1>...

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Chapter 4 Product spaces and independence 1. Product measures < 1 > Definition. Let X 1 ,..., X n be sets equipped with sigma-fields A 1 ,..., A n . The set of all ordered n -tuples ( x 1 ,..., x n ) , with x i X i for each i is denoted by X 1 × ... × X n or X i n X i . It is called the product of the { X i } . A set of the form A 1 × ... × A n = { ( x 1 ,..., x n ) X 1 × ... × X n : x i A i for each i } , with A i A i for each i , is called a measurable rectangle . The product sigma-field A 1 ... A n on X 1 × ... × X n is defined to be the sigma-field generated by all measurable rectangles. Remark. Even if n equals 2 and X 1 = X 2 = R , there is is no presumption that either A 1 or A 2 is an interval—a measurable rectangle might be composed of many disjoint pieces. The symbol in place of × is intended as a reminder that A 1 A 2 consists of more than the set of all measurable rectangles A 1 × A 2 . To keep the notation simple, I will mostly consider only measures on a product of two spaces, ( X , A ) and ( Y , B ) . Sometimes I will abbreviate symbols like M + ( X × Y , A B ) to M + ( X × Y ) , with the product sigma- fi eld assumed, or to M + ( A B ) , with the product space assumed. Similarly, M bdd ( A B ) will be an abbreviation for M bdd ( X × Y , A B ) , the vector space of all bounded, real-valued, product measurable functions on X × Y . Suppose µ is a fi nite measure on A and ν is a fi nite measure on B . The next theorem, which is usually called Fubini’s Theorem, asserts existence of a fi nite measure on A B whose integrals can be calculated by iterated integrals with respect to µ and ν . Remember the notation µ x h ( x , y ) for what would be written h ( x , y )µ( dx ) in traditional notation, the integral of h ( · , y ) with respect to µ with y held fi xed. < 2 > Theorem. For finite measures µ and ν , there is a uniquely determined finite measure on A B for which (i) ( A × B ) = A )(ν B ) for each measurable rectangle. Moreover, for each h in M bdd ( A B ) ,

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78 Chapter 4: Product spaces and independence (ii) the map x h ( x , y ) is A -measurable for each fixed y and the map y h ( x , y ) is B -measurable for each fixed x (iii) the map x ν y h ( x , y ) is A -measurable and the map y µ x h ( x , y ) is B -measurable (iv) µ x ( ν y h ( x , y ) ) = ν y ( µ x h ( x , y ) ) (v) the common value in (iv) is equal to h Remark. Properties (ii) and (iii) are necessary requirements for the iterated integrals in (iv) to make sense. Proof . The method of proof is a case study in the use of the generating class argument for λ -spaces, as developed in Section 2.11 . The main idea is to de fi ne the measure by means of the iterated integral Write H for the set of all functions h in M bdd ( A B ) for which (ii), (iii), and (iv) hold. It is very easy to check that H is a λ -space. For example, if h n H and h n h with h bounded then, by Monotone Convergence (for increasing sequences bounded from below by an integrable function), ν y h ( x , y ) = lim n →∞ ν y h n ( x , y ) , which establishes property (iii) for h . Similarly, four appeals to Monotone Convergence lead from the equality of iterated integrals for each h n
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