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 9 We quickly review the definition of measure zero. A set A R n is of measure zero if for every > 0, there exists a covering of A by rectangles Q 1 , Q 2 , Q 3 , . . . such that the total volume v ( Q i ) < . Remark. In this definition we can replace rectangles by open rectangles. To see this, given any > 0 let Q 1 , Q 2 , . . . be a cover of A with volume less than / 2. Next, choose Q i to be rectangles such that Int Q i Q i and v ( Q i ) < 2 v ( Q i ). Then Int Q 1 , Int Q 2 , . . . cover A and have total volume less than . We also review the three properties of measure zero that we mentioned last time, and we prove the third. 1. Let A, B R n and suppose B A . If A is of measure zero, then B is also of measure zero. 2. Let A i R n for i = 1 , 2 , 3 , . . . , and suppose the A i s are of measure zero. Then A i is also of measure zero. 3. Rectangles are not of measure zero. We prove the third property: Claim. If Q is a rectangle, then Q is not of measure zero. Proof. Choose < v ( Q ). Suppose Q 1 , Q 2 , . . . are rectangles such that the total volume is less than and such that Int Q 1 , Int Q 2 , . . . cover Q . The set Q is compact, so the HB Theorem implies that the collection of sets Int Q 1 , . . . , Int Q N cover Q for N suciently large. So, N Q Q i , (3.36) i =1 which implies that N v (...
View
Full
Document
This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.
 Fall '04
 unknown
 Angles

Click to edit the document details