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Unformatted text preview: Lecture 7 : Product Spaces 7.3 Product spaces and Fubinis Theorem Definition 7.3.1 If ( i , F i ) are measurable spaces, i I (index set), form Q i i . For simplicity, i = 1 . Q i i (write for this) is the space of all maps: I 1 . For Q i i , = ( i : i I, i i ) . is equipped with projections , X i : i , X i ( ) = i . [picture of square, 1 on one side, 2 on other, point in middle, maps under projection to each side] Definition 7.3.2 A product -field on is that generated by the projections: F = (( X i F i ) | F i F i ) . F 1 F 2 = ( | 1 F 1 2 F 2 ) F = ( X 1 F 1 ) ( X 2 F 2 ) . We now seek to construct the product measure . We start with the n = 2 case. ( , F ) = ( 1 , F 1 ) ( 2 , F 2 ). Suppose we have probability measures P i on ( i , F i ), i = 1 , 2. Then there is a natural way to construct P = P 1 P 2 (product measure) on ( , F ). Key idea: F 1 F 1 , F 2 F 2 , P ( F 1 F 2 ) = P 1 ( F 1 ) P 2 ( F 2 ). Theorem 7.3.3 (Existence of product measure and Fubini theorem) : (no- tation = ( 1 , 2 ) ) There is a unique probability measure P on ( 1 , F 1 ) ( 2 , F 2 ) such that for every non-negative product-measurable [measurable w.r.t. the productsuch that for every non-negative product-measurable [measurable w....
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This note was uploaded on 11/18/2009 for the course MATH 241 taught by Professor Reed during the Spring '09 term at Duke.
- Spring '09