Download Analysis of the phase space, asymptotic behavior and by Sköldstam 1975- Markus PDF

By Sköldstam 1975- Markus

ISBN-10: 9173739944

ISBN-13: 9789173739948

Research of the section house of heavy symmetric best (ISBN 9173739944)

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Extra info for Analysis of the phase space, asymptotic behavior and stability for heavy symmetric top and tippe top

Example text

Every Riemann integrable function is Lebesgue integrable and the integrals are equal. This fact is useful when a Lebesgue inif the function is in fact Riemann integrable, tegral needs to be evaluated then all of the theorems on Riemann integration can be used. Simple examples such as XQ show that there are Lebesgue integrable functions that are not Riemann integrable. In addition to integrating more functions, the Lebesgue integral overcomes two of the major deficiencies of the Riemann integral.

At this point, the Riemann and Lebesgue integrals are defined for bounded functions on an interval [a, b]. Every Riemann integrable function is Lebesgue integrable and the integrals are equal. This fact is useful when a Lebesgue inif the function is in fact Riemann integrable, tegral needs to be evaluated then all of the theorems on Riemann integration can be used. Simple examples such as XQ show that there are Lebesgue integrable functions that are not Riemann integrable. In addition to integrating more functions, the Lebesgue integral overcomes two of the major deficiencies of the Riemann integral.

D The function g has the same bound as f if f is bounded. There will be some instances when this fact will be important. An example to illustrate Lusin's Theorem may prove helpful. Let E C [a, b] be a perfect, nowhere dense set with positive measure such that a, b E E and consider the function XE . ) Write [a, b] - E as U (ak, bk) k=1 n and let Kn = [a, b] - U (ak, bk) Then each Kn is a finite union of disjoint closed . k=1 00 intervals, Kn+ 1 C Kn for each n, and E= n K, Let E > 0. By Theorem .

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