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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 H-B 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 (...
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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