This chapter revisits the Riemann integral in a way that sets up the motivation for Lebesgue–Stieltjes: Riemann integration is intuitive, but it fails exactly where modern analysis needs stability.

3.1 Definition of the integral

The Riemann integral is introduced via partitions and step-function bounds:

  • lower sums approximate from below,
  • upper sums approximate from above,
  • integrability means the gap between them can be made arbitrarily small.

A good conceptual point: the Riemann integral approximates the “area under the graph” by chopping the domain into subintervals.

3.2 Improper integrals

The chapter emphasizes how extending Riemann integration beyond closed bounded intervals requires extra conventions (limits at endpoints or infinity).

Two important warnings:

  • Continuity on a semiopen interval does not guarantee the improper integral exists.
  • Conditional convergence is fragile: ∫ f may converge while ∫ f diverges.

A canonical example is the oscillatory integral of sin(x)/x over an unbounded interval: it converges conditionally but is not absolutely convergent.

3.3 A nonintegrable function

Here the book highlights a function that defeats Riemann’s approach (the details vary by text, but the pattern is the same):

  • the function oscillates “too much” on every interval,
  • any domain-partition-based approximation cannot pin down a single limit.

This is the first punchline:

Riemann integration is sensitive to the behavior of a function on complicated sets, and it is not stable under the limiting operations that analysis wants to perform.

What changes in Lebesgue’s approach

Lebesgue integration flips the approximation viewpoint:

  • Instead of partitioning the domain (x-axis), it groups points by the values of the function (y-axis).
  • It replaces “length of subintervals” with a generalized notion of size: measure.

The rest of the book can be seen as designing the smallest amount of measure theory needed to make that shift practical.

Practice checklist

  • Work fluently with upper/lower sums and integrability criteria.
  • Classify improper integrals as absolute vs conditional.
  • Internalize why “domain partitions” can’t control highly discontinuous functions.