This book, first published in 2005, introduces measure and integration theory as it is needed in many parts of analysis and probability theory. The basic theory - measures, integrals, convergence theorems, Lp-spaces and multiple integrals - is explored in the first part of the book. The second part then uses the notion of martingales to develop the theory further, covering topics such as Jacobi's generalized transformation Theorem, the Radon-Nikodym theorem, Hardy-Littlewood maximal functions or general Fourier series. Undergraduate calculus and an introductory course on rigorous analysis are the only essential prerequisites, making this text suitable for both lecture courses and for self-study. Numerous illustrations and exercises are included and these are not merely drill problems but are there to consolidate what has already been learnt and to discover variants, sideways and extensions to the main material. Hints and solutions can be found on the author's website, which can be reached from www.cambridge.org/9780521615259. This book forms a sister volume to René Schilling's other book Counterexamples in Measure and Integration (www.cambridge.org/9781009001625).

'... thorough introduction to a wide variety of first year graduate level topics in analysis... accessible to anyone with a strong undergraduate background in calculus, linear algebra, and real analysis.'

Source: Zentralblatt MATH

'This is a concise and elementary introduction to measure and integration theory as need nowadays in many parts of analysis and probability theory.'

Source: L'Enseignement Mathématique

'I have not seen some of the topics that are mentioned above … treated successfully at undergraduate level before, and the book is worth having for these alone … [it] has the potential to revitalize the way that measure theory is taught.'

Source: Journal of the Royal Statistical Society