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 plurirectangle, 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
plurirectangle P1 contained in E and and a plurirectangle 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 plurirectangle (exercise) and since E ) P2 , %
P1 ) E % , we have that
%
# E = E...
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 Spring '14
 Angles, Ri, measure, Lebesgue measure, Meas

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