21-356 Lecture 29

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

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: 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...
View Full Document

This document was uploaded on 03/31/2014.

Ask a homework question - tutors are online