 
Summary: Contents
1. The Riemann and Lebesgue integrals. 1
2. The theory of the Lebesgue integral. 7
2.1. The Monotone Convergence Theorem. 7
2.2. Basic theory of Lebesgue integration. 9
2.3. Lebesgue measure. Sets of measure zero. 12
2.4. Nonmeasurable sets. 13
2.5. Lebesgue measurable sets and functions. 14
3. More on the Riemann integral. 18
3.1. The fundamental theorems of calculus. 21
3.2. Characterization of Riemann integrability. 22
1. The Riemann and Lebesgue integrals.
Fix a positive integer n. Recall that
Rn and Mn
are the family of rectangles in Rn
and the algebra of multirectangles in Rn
, respec
tively.
Definition 1.1. We let
F+
