A uniquely accessible book for general measure and integration,
emphasizing the real line, Euclidean space, and the underlying role
of translation in real analysis
"Measure and Integration: A Concise Introduction to Real
Analysis" presents the basic concepts and methods that are
important for successfully reading and understanding proofs.
Blending coverage of both fundamental and specialized topics, this
book serves as a practical and thorough introduction to measure and
integration, while also facilitating a basic understanding of real
analysis.
The author develops the theory of measure and integration on
abstract measure spaces with an emphasis of the real line and
Euclidean space. Additional topical coverage includes:
Measure spaces, outer measures, and extension theorems
Lebesgue measure on the line and in Euclidean space
Measurable functions, Egoroff's theorem, and Lusin's theorem
Convergence theorems for integrals
Product measures and Fubini's theorem
Differentiation theorems for functions of real variables
Decomposition theorems for signed measures
Absolute continuity and the Radon-Nikodym theorem
Lp spaces, continuous-function spaces, and duality theorems
Translation-invariant subspaces of L2 and applications
The book's presentation lays the foundation for further study of
functional analysis, harmonic analysis, and probability, and its
treatment of real analysis highlights the fundamental role of
translations. Each theorem is accompanied by opportunities to
employ the concept, as numerous exercises explore applications
including convolutions, Fourier transforms, and differentiation
across the integral sign.
Providing an efficient and readable treatment of this classical
subject, "Measure and Integration: A Concise Introduction to Real
Analysis" is a useful book for courses in real analysis at the
graduate level. It is also a valuable reference for practitioners
in the mathematical sciences.
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