hw4 - Tufts University Department of Mathematics Math 136...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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.

Ask a homework question - tutors are online