Lebesgue Integration and Measure
Lebesgue integration is a technique of great power and elegance which can be applied in situations where other methods of integration fail. It is now one of the standard tools of modern mathematics, and forms part of many undergraduate courses in pure mathematics. Dr Weir's book is aimed at the student who is meeting the Lebesgue integral for the first time. Defining the integral in terms of step functions provides an immediate link to elementary integration theory as taught in calculus courses. The more abstract concept of Lebesgue measure, which generalises the primitive notions of length, area and volume, is deduced later. The explanations are simple and detailed with particular stress on motivation. Over 250 exercises accompany the text and are grouped at the ends of the sections to which they relate; notes on the solutions are given.
Appendix apply assume Axiom of Completeness Baire class bounded interval Chapter characteristic functions closed interval compact containing continuous function converges almost everywhere Corollary countable cube decreasing deduce defined definition denote density function derivative disjoint intervals Dominated Convergence Theorem element elementary equation Euclidean example exists expressed finite measure follows Fourier series Fubini's Theorem function F given gives graph increasing sequence inequality integrable functions least upper bound Lebesgue integral Lebesgue measure Lemma linear mapping linear space linear subspace lower bound matrix Mean Value Theorem measurable function metric space Monotone Convergence Theorem non-empty norm notation null set one-one open intervals open sets open subset orthogonal orthonormal set period 2n positive proof of Theorem prove rational numbers real line real numbers result satisfies sequence of step sequence sn simple functions step functions Suppose tion topological union verify zero