This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nonnegative productmeasurable [measurable w.r.t. the productsuch that for every nonnegative productmeasurable [measurable w....
View
Full
Document
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
 Reed

Click to edit the document details