Best Introduction for the Financial Mathematician

This book is, as it were, manna from heaven for the aspiring financial mathematician, especially someone without a first degree in mathematics or a mathematically-based subject. The only prerequisites are a very good understanding of set-theory and some knowledge of the theory of continuous functions (at the level of a first course in real analysis, e.g. Apostol's Mathematical Analysis). The development is patient and there is sufficient help for the beginner (full solutions at the back, and, for practice, unproved propositions in the text with proofs at chapter-ends). The coverage is not overwhelming and anyone with the requisite preparation can digest the book in a term's work. Measure theory on its own is an incredibly dry subject. The authors do a great job of covering the essentials in about 300 pages, while making the subject interesting and applicable at the same time. A very attractive feature of the book is its brief focus on mathematical-finance applications. Most chapters end with a small section on such applications which is very useful for someone simultaneously studying mathematical finance. Particularly, it shows how to conceptualize financial models measure-theoretically. A very useful little volume indeed!

A solid introductory text

This text is a leisurely development of the major concepts in measure of probability theory. There are many useful examples that are sprinkled throughout the text that motivate the discussion. The solutions to the problems are also provided. This book is suitable for advanced undergraduate. Although I don't have any major complaints about the book, there already exists a much better book with many more exercises, diagrams, and much more thorough development and extension of the principal concepts. That book is Jones' "Lebesgue integration on Euclidean space", which I cannot recommend highly enough.

Highly accessible

Highly accessible and clear intro to measure and Lebesgue integration. Can relax with this book while waiting for the train after work. Only minor negatives are: (1) not enough exercises and (2) there are typos but Springer Verlag doesn't provide any errata list whatsoever.

(Update 2008) There are now other similar books on the market and the paucity of exercises in this book is just not acceptable. For an inexpensive alternative, I recommend Klambauer's "Real Analysis" published by Dover. He uses Carathéodory's definition of a measurable set which (to me) is a faster path to results. Also his exercises are insightful.