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21-356 Lecture 30

It follows by the previous theorem that e is

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Unformatted text preview: = 0. It follows by the previous theorem that # E is Peano—Jordan measurable with measure zero. Step 2: Assume that # E ) RN is Peano—Jordan measurable with measure zero. Since E is bounded, so is E and so there exists a rectangle R containing # E . Since meas # E = 0 by the previous theorem there exists a a pluri-rectangle P containing # E such that meas P % (. 78 The set R \ P is a pluri-interval and thus we can write it as disjoint unions of rectangles, n E R\P = Ri . i=1 Let P1 be the pluri-interval given by the union of all the rectangles Ri that are contained in E , so that P1 ) E . Let P2 := P - P1 . We claim that E ) P2 . Fix x ! E . If x does not belong to P2 , then in particular it cannot belong to P and so it belongs to R \ P . Hence, there exists Ri such that x ! Ri . But then Ri must be contained in E . Indeed, if not, then there exists y ! Ri + (R \ E ). It follows by Exercise 139 that the segment S joining x and y must contained a point on the boundary of # E , which is a contradiction since the segment S is contained in Ri and Ri does not intersect P...
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