Unformatted text preview: P ) .
Taking the supremum over all partitions P of R, we get
A
inf {meas P : P plurirectangle, 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 plurirectangle
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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 Spring '14
 Angles, Ri, measure, Lebesgue measure, Meas

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