Download e-book for iPad: An introduction to Lebesgue integration and Fourier series by Howard J. Wilcox

By Howard J. Wilcox

ISBN-10: 0882756141

ISBN-13: 9780882756141

Undergraduate-level advent to Riemann essential, measurable units, measurable services, Lebesgue critical, different themes. a number of examples and routines.

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Additional info for An introduction to Lebesgue integration and Fourier series

Example text

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)

Let e > 0, and let G be open in [c,d] with ( [c,d] \A ) C G and m*(G) < m *( [c,d] \A) + e. Then H = G U [a,c) is open in [a,b] , and ( [a,b] \A) C H, so m*( [a,b ] \A) < m *(G U [a,c)) = m*(G) + (c - a) < m*( [c,d] \A) + (c - a) + e. Since e was arbitrary, m*([a,b]\A) < m*([c,d]\A) + (c - a). On the other hand, if G ' is open in [a,b] , G ' ::) ( [a,b] \A), and m*(G ' ) < m * ([a,b] \A) + e, then [a,c) C G ', H ' = G '\ [a,c) is open in [c,d] , contains [c,d) \A , and m*([c,d]\A ) < m*(H ' ) = m*(G ') - (c - a) < m*([a,b]\A) - (c - a) + e.

B). Prove that if IJ. (Q) = 0. 4 ES). (a) (b) (c) Prove that S is closed under countable intersections. Prove that if A, B E S, then A \B E S. Show that if 1J. (A n B). 5 Prove that the set function 1J. 2 is a measure. 6 Prove that the "trivial measure" of Example 6 . 4 is not closed under countable unions. Show however that 1J. satisfies (a), (b), (c) of Definition 6. 1 , and that 1J. is consistent with lengths of intervals. 5 are equivalent. Show that "Jordan content" is consistent with lengths of intervals.

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


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