Up to Chapter 5, the Lebesgue–Stieltjes integral is a definition with good abstract properties. Chapter 6 turns it into a computational tool: how do you actually evaluate integrals and do calculus with them?

6.1 Evaluation of integrals

The practical theme is that Stieltjes measures can have both:

  • a continuous part (like ordinary length), and
  • an atomic/jump part (point masses).

So computations often split into:

1) an ordinary integral on intervals where α behaves smoothly, and 2) explicit “point contributions” at discontinuities of α.

A typical rule of thumb:

  • On open intervals where α is constant, the integral is zero.
  • At a jump point a, the integral over the single-point interval contributes f(a) · (α(a+) − α(a−)).

This is the Stieltjes analog of “probability mass at a point.”

6.2 Two theorems of integral calculus

This section is where you recognize familiar calculus laws in Stieltjes form.

Expect versions of:

  • change of variable (substitution)
  • integration by parts

…but with extra bookkeeping for jumps and one-sided behavior.

If you think probabilistically: integration by parts is the engine behind many expectation identities, and change of variable is the engine behind distribution transforms.

6.3 Integration and differentiation

Here the book connects Lebesgue–Stieltjes integration to derivatives and to “differentiate under the integral sign” arguments.

The key message is: differentiation interacts nicely with the integral when you have control (uniform bounds, domination, or regularity of the integrand in parameters).

This is where Lebesgue theory becomes dramatically more pleasant than Riemann theory: you can justify interchange of limit processes with integrals under precise hypotheses.

Mental model

When α is smooth, Lebesgue–Stieltjes calculus looks like ordinary calculus.

When α has jumps, you get a hybrid calculus:

  • continuous integration + discrete summation.

That hybrid viewpoint is exactly what you need for real-world models (e.g., mixed distributions).

Practice checklist

  • For a step-like α, practice turning ∫ f dα into a finite sum of point masses.
  • For a differentiable α, practice converting ∫ f dα into an ordinary integral involving α′.
  • Keep a habit: always check where α is discontinuous, and treat those points separately.