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21-356 Lecture 29

# E let y rn e prove that the segment s joining x and

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Unformatted text preview: let x ! E , let y ! RN \ E . Prove that the segment S joining x and y intersects # E . Theorem 140 A bounded set E ) RN is Peano—Jordan measurable if and only if its boundary is Peano—Jordan measurable and it has Peano—Jordan measure zero. Proof. We begin by observing that if P is a pluri-rectangle, then meas # P = 0 (why?), and so meas P = meas P = meas P % . Step 1: Assume that E ) RN is Peano—Jordan measurable and let R be a rectangle containing E . By the previous theorem, for every ( > 0 there exist a pluri-rectangle P1 contained in E and and a pluri-rectangle P2 is containing E such that 0 % meas P2 ' meas P1 % (. Hence, ' ( % % meas P2 \ P1 = meas P2 ' meas P1 = meas P2 ' meas P1 % (. % Note that P2 \ P1 is still a pluri-rectangle (exercise) and since E ) P2 , % P1 ) E % , we have that % # E = E...
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