This chapter builds the “analysis toolkit” used throughout Lebesgue–Stieltjes integration: limits, monotonicity, step functions, and regularity conditions like bounded variation.

2.1 Monotone sequences

Monotone sequences are the simplest vehicle for limits.

  • An increasing sequence that is bounded above converges.
  • A decreasing sequence that is bounded below converges.

These facts let you replace messy limiting arguments with monotone bounding constructions—exactly what happens later when defining the integral by approximation.

2.2 Double series

Double sums show up when you decompose a function into a series of simple (step-like) components.

Key idea: switching the order of summation is not always harmless unless you have absolute convergence or another controlling condition.

This “order matters unless controlled” principle foreshadows why measure theory cares so much about nonnegativity and domination.

2.3 One-sided limits

Lebesgue–Stieltjes theory works naturally with one-sided limits:

  • f(t−) and f(t+)

because the measure generated by a monotone function α can jump, and endpoints of intervals are asymmetric in Stieltjes-style constructions.

2.4 Monotone functions

Monotone functions have strong structure:

  • one-sided limits exist everywhere inside the interval,
  • discontinuities are well behaved (they cannot oscillate arbitrarily).

Later, α itself is monotone increasing; these properties are how you keep the induced measure consistent.

2.5 Step functions

Step functions are the “atoms” for the construction.

  • They are easy to integrate.
  • They approximate more complicated functions.

Most proofs in the early Lebesgue development are: prove something for step functions → extend by approximation.

2.6 Positive and negative parts

Decompose any real-valued function into

  • f = f⁺ − f⁻, with f⁺ = max(f, 0), f⁻ = max(−f, 0)

This is a standard trick because the integral is first built for nonnegative functions, then extended by linearity to signed functions via f⁺ and f⁻.

2.7 Bounded variation and absolute continuity

Two regularity classes are highlighted because they interact beautifully with Stieltjes measures.

Bounded variation (BV)

A function has bounded variation on an interval if the total variation (sup over all partitions of the sum of absolute increments) is finite.

Why BV matters:

  • BV functions have one-sided limits everywhere.
  • Any BV function can be expressed (non-uniquely) as the difference of two monotone increasing functions.

This decomposition is structural gold: it lets you reduce “BV-generated measures” to “monotone-generated measures.”

Absolute continuity (AC)

Absolute continuity is a stronger condition than continuity. Intuitively:

  • Small total length of disjoint subintervals ⇒ small total oscillation of f over those intervals.

A key direction proved in the book is:

  • AC ⇒ BV on a finite interval.

Why AC matters later: it is the condition under which Stieltjes integration connects tightly to classical differentiation and to change-of-variables style results.

Mental model for later chapters

If Chapter 1 is about “the stage” (ℝ and completeness), Chapter 2 is about “the actors”:

  • monotone objects,
  • step approximations,
  • variation controls.

Lebesgue–Stieltjes integration is engineered to be stable under limits, and these preliminaries are the stability toolkit.

Practice checklist

  • Prove one-sided limits exist for monotone functions.
  • Compute variation for simple examples; build intuition for BV vs non-BV.
  • Practice f = f⁺ − f⁻ decompositions.