Replace the Riemann integral with the calculus integral
and the Newton integral only [Chapter 8].
• [Medium drip]
Add sufficient elements of the theory to give a rudimentary
picture of the integral and its properties [Chapters 9 and 10].
This is easier than the usual introductions to the Riemann integral,
but it happens to include the calculus integral, the Newton integral,
and the Lebesgue integral.
• [Strong drip]
Throw in the basics of zero measure, the Mini-Vitali theorem, and functions of bounded variation
[Chapters 11, 12, and 13]. While still at an elementary level, this allows
a full treatment of the Lebesgue differentiation theorem, absolute
continuity, and the most general version of the fundamental theorem
of the calculus.
• [Full drip]
Add parts of the advanced material as desired [Chapters
15, 18, and 17].T his brings the student up to a serious level in
integration theory but not in the usual direction.The culmination, in
Chapter 17, is the Lebesgue program for the measure-theoretic development
of his integral.The standard approach starts with measure
theory and slowly (some say painfully) develops the integral and its
theory; this easier version starts with the integral and the measure
theory develops naturally from it.