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

ISBN-10: 0882756141

ISBN-13: 9780882756141

Undergraduate-level creation to Riemann vital, measurable units, measurable services, Lebesgue critical, different themes. various examples and routines.

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Additional info for An introduction to Lebesgue integration and Fourier series

Sample text

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,)

F(x) > a } = n9 t {xlfn(x) > a} . 0 Under the conditions of the proposition, g(x) = nlim fn(x) +• and h(x) = lim fn(x) are measurable. 8 Corollary: Proof: Use the relationship g(x) = lim (Iub UnCx)l n k +• a similar relationship for h(x). > k}), and 0 1 9. Simple F unctions In discussing the Lebesgue Integral in the next chapter, we will have need of some particularly uncomplicated measurable functions. These will play a role analogous to that of step functions in the Riemann theory. 1 Definition : A simple function finitely many values.

Therefore, the calculation in the proof of the Theorem shows that l/11(x) -f(x) I < 1 /n for all x E A . Unifonnity follows. 35. 0 20. 1 Prove ( 3) '* ( 4) in Proposition 1 7. 2. 2 Prove ( 4) => ( 1 ) in Proposition 1 7. 2. 3 Prove the remainder of Proposition 1 7. 2. 4 (a) Prove that if f is measurable, then {x E A I c = f(x)} is measurable for each real number c. (b) Show that the converse of (a) is false if m(A ) > 0. (Hint : by Exercise 1 6 . ) = 0, then every function is measurable on A . 6 If B is any set, f:B _.