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by: Jayde Lang

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# INTRODUCTION TO ANALYSIS I MATH 321

Marketplace > Rice University > Mathematics (M) > MATH 321 > INTRODUCTION TO ANALYSIS I
Jayde Lang
Rice University
GPA 3.86

Stephen Semmes

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COURSE
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Stephen Semmes
TYPE
Class Notes
PAGES
1
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 1 page Class Notes was uploaded by Jayde Lang on Monday October 19, 2015. The Class Notes belongs to MATH 321 at Rice University taught by Stephen Semmes in Fall. Since its upload, it has received 13 views. For similar materials see /class/224928/math-321-rice-university in Mathematics (M) at Rice University.

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Date Created: 10/19/15
Math 321 Introductory Lebesgue Measure Handout 3a A set E Q R has Lebesgue measure 0 if and only if ME 0 Similarly7 E has measure 0 in the restricted sense using coverings by nitely many open intervals if and only if ME 0 For any E Q R7 ME lt 00 if and only if E is bounded However7 unbounded sets may have nite Lebesgue outer measure7 since countable sets automatically have Lebesgue measure 0 lfAQEQR7then MA lt ME 1 and MA S ME 2 This is because coverings of E by nitely or countably many open intervals are also coverings of A of the same type Thus the in ma in the de nitions of MA and MA are taken over wider classes of sums than the in ma in the de nitions of ME and ME7 respectively Of course7 a covering of A is not necessarily a covering of E7 and the preceding inequalities may be strict For each E Q R7 ME ME 3 More precisely7 ME 3 ME since E Q E7 and the analogous statement for A instead of M also holds for the same reason To get the opposite inequality7 let 177In be nitely many open intervals such that E Q UL Ii7 which implies that E Q ULl By expanding these intervals slightly7 we can get open intervals 17 7 such that E Q ULl I and 21 is arbitrarily close to ELI Thus the sums in the de nition of ME can be approximated by the sums in the de nition of ME7 as desired

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