This book presents a unified treatise of the theory of measure and
integration. In the setting of a general measure space, every concept is
defined precisely and every theorem is presented with a clear and
complete proof with all the relevant details. Counter-examples are
provided to show that certain conditions in the hypothesis of a theorem
cannot be simply dropped. The dependence of a theorem on earlier
theorems is explicitly indicated in the proof, not only to facilitate
reading but also to delineate the structure of the theory. The precision
and clarity of presentation make the book an ideal textbook for a
graduate course in real analysis while the wealth of topics treated also
make the book a valuable reference work for mathematicians.
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This book is meant as a text for a first-year graduate course in
analysis. In a sense, it covers the same topics as elementary calculus
but treats them in a manner suitable for people who will be using it in
further mathematical investigations. The organization avoids long chains
of logical interdependence, so that chapters are mostly independent.
This allows a course to omit material from some chapters without
compromising the exposition of material from later chapters.
This book presents a unified treatise of the theory of measure and
integration. In the setting of a general measure space, every concept is
defined precisely and every theorem is presented with a clear and
complete proof with all the relevant details. Counter-examples are
provided to show that certain conditions in the hypothesis of a theorem
cannot be simply dropped. The dependence of a theorem on earlier
theorems is explicitly indicated in the proof, not only to facilitate
reading but also to delineate the structure of the theory. The precision
and clarity of presentation make the book an ideal textbook for a
graduate course in real analysis while the wealth of topics treated also
make the book a valuable reference work for mathematicians.
This material is intended to contribute to a wider appreciation of the mathematical words "continuity and linearity". The book's purpose is to illuminate the meanings of these words and their relation to each other.