Lecture7

Lecture7 - Lecture 7 : Product Spaces 7.3 Product spaces...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the 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 non-negative product-measurable [measurable w.r.t. the productsuch that for every non-negative product-measurable [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.

Page1 / 4

Lecture7 - Lecture 7 : Product Spaces 7.3 Product spaces...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online