Unformatted text preview: Tufts University Department of Mathematics Math 136 Homework 4 Due, Tuesday, March 4, 2008 in class. Here is a theorem that may be used on this homework even if we don’t prove it until after this homework is due. Theorem 1 Let A ⊂ R n . Then A has measure zero if and only if for each > , there is a countable collection of open rectangles (with sides parallel the axes) { U j j ∈ N } so that A ⊂ [ j ∈ N U j and ∞ X j =1 v ( U j ) < . 1. (20 points) You will show that open rectangles have volume and the volume of an open rectangle is the same as the volume of its closure, a closed rectangle. (a) Let a ∈ R and b ∈ R with a < b . Show using the definition of integral and a good partition that R 1 [ a,b ] = b a . This shows that the volume of [ a,b ] as a set is the same as the volume of [ a,b ] as an interval. (b) Let a ∈ R and b ∈ R with a < b . Prove v (( a,b )) = v ([ a,b ]) using the limit theorem for the integral and the partitions P n = [ a,a + 1 /n ] , [ a + 1 /n,b...
View
Full
Document
This note was uploaded on 03/27/2008 for the course MATH 136 taught by Professor Quinto during the Spring '08 term at Tufts.
 Spring '08
 Quinto
 Math

Click to edit the document details