Lebesgue Integration and Measure 1st Edition by Alan J. Weir (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐At first glance the book seems to be great. It’s something less than a textbook, making it more approachable to the “beginner” (as the book calls it). One must wonder, however, what exactly the beginner wants.One thing the beginner apparently does NOT want is mathematical rigor, which the book applies inconsistently. Some basic theorems are given pages upon pages of proof, while more difficult theorems are not only not proven (the author recommends a “classical textbook” for the “experienced reader”) but not even stated specifically. Furthermore several of the proofs that are supplied (which are probably assumed to be “classical” and therefore not edited) are actually flawed, relying on circular logic or jumps which are not actually logically valid.The last qualm I have with the book is that they are so lax with notation that many theorems appear to mean something completely other than what they appear to (especially concerning higher-dimensional spaces). This may be a product of this book being essentially unchanged in thirty years.After all these complaints I will say that when the book gets it right, as it often does, it is easy to read and understandable. So long as it is not examined too closely.

⭐For an intro – an understandable intro – to Lebesgue integration this is one of the best books you can buy.

⭐This book introduces Lebesgue Measure in the simplest possible way. Really a must for anyone’s library! Every Grad student will benefit from this text.

⭐Its a very good text for a first meeting on Lebesgue integration, measure and functional analysis. Rigorous, elegant and simple. A quality book for pure and applied mathematics.However I found two little mistakes:In page 151, in the proof of the integral of a transformation, he makes use of the Dominated Convergence Theorem two times (one first time, at the begining of page 151, is right). Thats wrong because we can’t “dominate” the function “g” when K -> inf. The correct proof involves divide the function in positive and negative parts and then aplicate Monotone Convergence Theorem. The same in the end of the proof when he generalizes to infinite measure sets.In page 157, equation (7) should be verified when ||h||<2*delta, not ||h||