An introduction to Lebesgue integration and Fourier series by Howard J. Wilcox

By Howard J. Wilcox

Undergraduate-level creation to Riemann necessary, measurable units, measurable services, Lebesgue necessary, different issues. a number of examples and routines.

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Example text

Is a collection of subsets ofE. ). Proof: We present the proof in several parts. (1) Finite collections ofopen intervals. It is obvious that if / 1 and /2 are open intervals in E, then m*(/ 1 U /2 ) < m*{I1 ) + m*(/2 ) . ) It follows that = for any finite collection of open intervals. ) (2) Countable collections ofopen intervals. Given nU= 1 I,. 's are disjoint open intervals (Theorem? 2). ) < e. ) + e. • • For each n = 1 ,2, . . ,N, there is an open interval K,. K,. ] C J,. and such that 24 L E B ESGUE I NT E G R AT I ON A N D F OU R I E R SE R I ES m*(J,)

1 2 Let C be the Cantor set (Example l 2. 2 1 ). Let D C [ 0, 1 ] be a nowhere dense measurable set with m(D) > 0 (Exercise 1 6. 22). 29). At each stage of the construction of C and of D, a certain finite number of open intervals of [ 0, 1 ] are deleted (put into [ 0, 1 ] \C or [ 0, 1 ] \D). Let g map the in­ tervals put into [ 0, 1 ] \D at the nth stage linearly onto the intervals put into [ 0, 1 ] \c at the nth stage, for n = 1 ,2, . . ) Thus g is monotone and defined at every element of [ 0, 1 ] \D, m apping onto [ 0, 1 ] \C.

1 we proved that if G1 and G2 are open in E, then m*(G 1) + m*(G2) > m*(G 1 U G2) + m*(G 1 n G2). 3) and we use this relationship to prove the following useful criterion for a set to be measurable. 2 Theorem : A m(G 1 n G2) 0 be given. Then there exist open G. 4)+e/2. Now m(G 1 n G2) =m(G 1 )+m(G2)-m(G 1 U G2)

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