New PDF release: An introduction to Lebesgue integration and Fourier series

By Howard J. Wilcox

ISBN-10: 0882756141

ISBN-13: 9780882756141

Undergraduate-level advent to Riemann vital, measurable units, measurable services, Lebesgue crucial, different issues. a number of examples and routines.

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Extra info for An introduction to Lebesgue integration and Fourier series

Example text

One might reasonably ask whether M or some other large class of sets on which m * is a measure could be more easily defined. We will present one such large class (the Borel Sets) and then show that the class M of measurable sets has many more sets than the class of Borel Sets. First we need the following concept. 1 Definition : Let S be a collection of subsets of E. Then S is called a countably-additive class of sets (or a a-algebra) if (1) E E S. (2) A-E S ,. 4 E S. (3) ;Y A ; E S whenever {A; } ;:, 1 c S.

A1a2a 3 �2!.. ; ; , where the " " ) = expression on the left is ternary with no 1 's, and the expression on the right is binary. Prove that f is onto but not 1 - 1 . ( See Exercise 1 6 . 19 Prove that the function f in Exercise • • • • • • 1 6. 1 8 is continuous. 20 Prove that C is uncountable. (Hint : use the ternary characterization of C, and a modification of the usual diagonal argument used to show that (0, 1 ) is uncountable. ). Prove that every point of C is a limit point of C ; that is, for every E C and every € > 0, there is b E C with - b €.

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. 2 Proposition : A function g :A -+ tR numbers b 1 , • • • , b n and pairwise n n with A = i=ul B1 and g = i=l:l btXs ·· I -+ tR is a measurable function with is simple if and only if there are real disjoint measurable sets B�o .

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


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