Meant for advanced undergraduate and graduate students in
mathematics, this lively introduction to measure theory and
Lebesgue integration is rooted in and motivated by the historical
questions that led to its development. The author stresses the
original purpose of the definitions and theorems and highlights
some of the difficulties that were encountered as these ideas were
refined. The story begins with Riemann's definition of the
integral, a definition created so that he could understand how
broadly one could define a function and yet have it be integrable.
The reader then follows the efforts of many mathematicians who
wrestled with the difficulties inherent in the Riemann integral,
leading to the work in the late 19th and early 20th centuries of
Jordan, Borel, and Lebesgue, who finally broke with Riemann's
definition. Ushering in a new way of understanding integration,
they opened the door to fresh and productive approaches to many of
the previously intractable problems of analysis.
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