The epilogue steps back and answers a natural question:

If Lebesgue (and Lebesgue–Stieltjes) integration is so powerful, why do people still invent new integrals?

10.1 The Lebesgue integral and its successors

Lebesgue integration is a foundation, but it is not the end of the story.

Some later integrals were designed to capture functions that are not Lebesgue integrable while preserving certain calculus identities.

A recurring tension:

  • Generality: integrate more functions.
  • Structure: keep familiar theorems (FTC, substitution, etc.).

Pushing one often compromises the other.

10.2 “Riemann strikes back”

The chapter discusses integrals that look more “Riemann-like” (partition-based) but are powerful enough to integrate broader classes than the classical Riemann integral.

The general pattern is:

  • allow partitions that adapt to the function (gauge-like refinement rules),
  • regain strong versions of the Fundamental Theorem of Calculus,
  • integrate some derivatives that are not Lebesgue integrable.

These integrals are relevant when your goal is not measure theory but rather calculus restoration.

How to place Lebesgue–Stieltjes in the bigger map

My takeaway from the book’s arc:

  • Use Lebesgue–Stieltjes when you care about convergence, probability, and functional-analytic structure.
  • Look at “successor integrals” when you specifically need stronger calculus identities for pathological objects.

For engineering and applied work, Lebesgue–Stieltjes is often the sweet spot: powerful enough for real analysis and probability, yet concrete enough to compute with (especially when α has a probabilistic interpretation).

Practice checklist

  • Be able to explain why Lebesgue beats Riemann (convergence + null sets).
  • Be able to explain why some gauge integrals beat Lebesgue for derivatives.
  • Keep a mental separation between: “integration as area/measure” vs “integration as inverse differentiation.”