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

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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 > 0 , 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 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 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 - 1 /n ] , [ b - 1 /n, b ] for n > 2 / ( b - a ). Note that this partition is especially fine near where 1 ( a,b ) is discontinuous.
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