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

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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 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 intervala 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, real-valued, product measurable functions on X Y . Suppose is a f nite measure on A and is a f nite measure on B . The next theorem, which is usually called Fubinis Theorem, asserts existence of a f nite measure on A B whose integrals can be calculated by iterated integrals with respect to and . Remember the notation x h ( x , y ) for what would be written R h ( x , y )( dx ) in traditional notation, the integral of h ( , y ) with respect to with y held f xed. < 2 > Theorem. For finite measures and , there is a uniquely determined finite measure on A B for which (i) 0( A B ) = ( A )( 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 A-measurable for each fixed y and the map y 7 h ( x , y ) is B-measurable for each fixed x (iii) the map x 7 y h ( x , y ) is A-measurable and the map y 7 x h ( x , y ) is B-measurable (iv) x ( y h ( x , y ) ) = y ( 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 -spaces, 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 -space. 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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Product - Chapter 4 Product spaces and independence 1....

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