21-356 Lecture 29

Taking the supremum over all partitions p of r we get

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Unformatted text preview: P ) . Taking the supremum over all partitions P of R, we get A inf {meas P : P pluri-rectangle, E ) P } % /E (x) dx. R Remark 138 In view of the previous theorem, if a bounded set E ) RN has Peano—Jordan measure zero, then for every ( > 0 there exists a pluri-rectangle P is containing E such that meas P % (. Hn By writing P as a union of disjoint rectangles, P = i=1 Ri , we have that n + i=1 meas Ri % (. This implies that E has Lebesgue measure zero. However, the opposite is not true. For example the set E := [0, 1] + Q has Lebesgue measure zero (since it is countable), but it is not Peano Jordan measurable and its outer measure is actually one. Indeed, its characteristic function is the Dirichlet function. 77 Wednesday, March 30, 2011 Exercise 139 Let E ) RN ,...
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