Measure, Integration & Real Analysis, 2021a

Chapter 3
Integration
To remedy deficiencies of Riemann integration that were discussed in Section 1B,
in the last chapter we developed measure theory as an extension of the notion of the
length of an interval. Having proved the fundamental results about measures, we are
now ready to use measures to develop integration with respect to a measure.
As we will see, this new method of integration fixes many of the problems with
Riemann integration. In particular, we will develop good theorems for interchanging
limits and integrals.
Statue in Milan of Maria Gaetana Agnesi,
who in 1748 published one of the first calculus textbooks.
A translation of her book into English was published in 1801.
In this chapter, we develop a method of integration more powerful
than methods contemplated by the pioneers of calculus.
©Giovanni Dall’Orto
Measure, Integration & Real Analysis, by Sheldon Axler
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