This book is both a text and a paean to twentieth-century real
variables, measure theory, and integration theory. As a text, the book
is aimed at graduate students. As an exposition, extolling this area of
analysis, the book is necessarily limited in scope and perhaps
unnecessarily unlimited in id- syncrasy. More than half of this book is
a fundamental graduate real variables course as we now teach it. Since
there are excellent textbooks that generally cover the course material
herein, part of this Preface renders an apologia for our content,
presentation, and existence. The following section presents our syllabus
properly liberated from too many demands. Subsequent sections
dealwithoutline, theme, features, andtherolesofFourieranalysisandVitali,
respectively. Mathematics is a creativeadventure drivenby beauty,
structure, intrinsic mathematicalproblems,
extrinsicproblemsfromengineeringandthesciences, and serendipity. This
book treats integration theory and its fascinating creation through the
past century. What about the rest of our title? "Analysis"is many
subjects to many mathematicians."Modern analysis" is hardly a constraint
for a single volume such as ours; one can argue the opposite.
Di?erentiation and integrationare still the essence of analysis, and,
along with "integration", the title could very well have included the
word "di?erentiation" because of our emphasis on it. Guided by the
creativity of mathematics,
ourtitleismeanttoassertthatthetechnologywehaverecorded is a basis for
many of the analytic adventures of our time. Syllabus We shall outline
the material we have used in teaching a ?rst-year graduate course in
real analysis. Sometimes a student will take only the ?rst semester of
this two-semester sequence.
