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Unformatted text preview: Chapter 4 Product spaces and independence 1. Product measures < 1 > Definition. Let X 1 ,..., X n be sets equipped with sigmafields A 1 ,..., A n . The set of all ordered ntuples ( 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 sigmafield A 1 ⊗ ... ⊗ A n on X 1 × ... × X n is defined to be the sigmafield 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 f 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, realvalued, product measurable functions on X × Y . Suppose µ is a f nite measure on A and ν is a f nite measure on B . The next theorem, which is usually called Fubini’s Theorem, asserts existence of a f 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 R h ( x , y )µ( dx ) in traditional notation, the integral of h ( · , y ) with respect to µ with y held f xed. < 2 > Theorem. For finite measures µ and ν , there is a uniquely determined finite measure on A ⊗ B for which (i) 0( A × B ) = (µ A )(ν B ) for each measurable rectangle. Moreover, for each h in M bdd ( A ⊗ B ) , 78 Chapter 4: Product spaces and independence (ii) the map x 7→ h ( x , y ) is Ameasurable for each fixed y and the map y 7→ h ( x , y ) is Bmeasurable for each fixed x (iii) the map x 7→ ν y h ( x , y ) is Ameasurable and the map y 7→ µ x h ( x , y ) is Bmeasurable (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 λspaces, as developed in Section 2.11. The main idea is to de f 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 λspace. For example, if h n ∈ H and h n ↑ h with h bounded then, by Monotone Convergence (for increasing sequences bounded...
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This note was uploaded on 11/21/2009 for the course STAT 330 taught by Professor Davidpollard during the Spring '09 term at Yale.
 Spring '09
 DavidPollard
 Probability

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