Chapter 4 is the core construction: it defines a measure on the real line from a monotone function α, then defines the integral by approximating functions with step-like building blocks.

4.1 Measure of an interval

The Stieltjes viewpoint is: instead of measuring intervals only by length, measure them by how much α increases.

  • Think of α as a “cumulative distribution” (it can have jumps and flat parts).
  • The measure of an interval is built from increments of α with careful attention to endpoints.

This is the minimal bridge from classical analysis (monotone functions and one-sided limits) to measure.

4.2 Probability measures

A major motivation for Stieltjes integration is probability.

  • If α is a cumulative distribution function (CDF), the measure induced by α is exactly the probability measure.
  • Lebesgue–Stieltjes integration then becomes “expected value” in a unified way that includes discrete, continuous, and mixed distributions.

4.3 Simple sets

To build a full measure theory without drowning in abstraction, the book introduces a practical class of “simple sets” and shows how to measure them consistently.

This mirrors how step functions are used as the simple class for functions.

4.4 Step functions revisited

Step functions are reintroduced in a measure-aware way:

  • A step function is a finite linear combination of indicator functions of intervals.
  • Its integral is “sum of heights × measure of supporting intervals.”

Once step functions are integrated, everything else is approximation.

4.5 Definition of the integral

The Lebesgue–Stieltjes integral is defined first for nonnegative measurable functions.

Operationally:

  1. approximate f from below by simple/step-like functions,
  2. define the integral as a supremum of the integrals of those approximants,
  3. extend to signed functions via f = f⁺ − f⁻ (when both parts are integrable).

What’s new compared to Riemann:

  • the “approximants” are more flexible than a single step function tied to one partition,
  • and the notion of “size” is measure-based, not length-based.

4.6 The Lebesgue integral as a special case

If you choose α(x) = x, the induced measure is ordinary length and the Lebesgue–Stieltjes integral becomes the Lebesgue integral.

Two practical cross-checks emphasized in the book:

  • If f is Riemann integrable on a closed interval, then it is also Lebesgue integrable there and the values agree.
  • The equivalence does not automatically extend to improper Riemann integrals (conditional convergence can break Lebesgue integrability).

Mental model

You can summarize the construction like this:

  • α determines what it means for a set to be small.
  • Step/simple functions provide controllable approximations.
  • The integral is a limit of these approximations, chosen so that key limit theorems become true.

Practice checklist

  • Compute measures of intervals for simple α (purely continuous, purely jump, mixed).
  • Integrate step functions against α.
  • Verify how α(x)=x collapses the theory back to standard Lebesgue integration.