Lebesgue Integration and Measure

A textbook for the undergraduate who is meeting the Lebesgue integral for the first time, relating it to the calculus and exploring its properties before deducing the consequent notions of measurable functions and measure.A textbook for the undergraduate who is meeting the Lebesgue integral for the first time, relating it to the calculus and exploring its properties before deducing the consequent notions of measurable functions and measure.Lebesgue integration is a technique of great power and more >>

A textbook for the undergraduate who is meeting the Lebesgue integral for the first time, relating it to the calculus and exploring its properties before deducing the consequent notions of measurable functions and measure.A textbook for the undergraduate who is meeting the Lebesgue integral for the first time, relating it to the calculus and exploring its properties before deducing the consequent notions of measurable functions 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 Weirýýýs 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.Preface; 1. The completeness of the reals; 2. Null sets; 3. The Lebesgue integral on R; 4. The Lebesgue integral on Rk; 5. The convergence theorems; 6. Measurable functions and Lebesgue measure; 7. The spaces Lp; Appendix: the elements of topology; Solutions; References; Index.ýýýThe book is easy to read, partly because of the treatment adopted, and partly because of the quality of the exposition. Dr Weirýýýs style is clear, friendly and informal; he shows how the results fit in with the readerýýýs intuition; he highlights the important things and warns of the difficult things (these warnings when a hard bit is coming up are most confidence-preserving). He does not aim at maximum generality at the cost of understanding. The examples are chosen with care, many of them being, in effect, lemmas that will be needed later in the proofs of theorems.ýýý Mathematical Gazette << less