After the construction, Chapter 5 justifies why this integral is worth the effort: it behaves well under limits and it ignores “negligible” changes.

5.1 Basic properties

The integral satisfies the properties you expect from any sane linear functional:

  • Linearity: ∫(af + bg) = a∫f + b∫g (when defined).
  • Monotonicity: f ≤ g ⇒ ∫f ≤ ∫g.
  • Positivity: f ≥ 0 ⇒ ∫f ≥ 0.
  • Triangle-type bounds: ∫f ≤ ∫ f .

The point is not that these are surprising; it’s that they are stable under the generalized definition (via approximation).

5.2 Null functions and null sets

This section formalizes the “ignore negligible sets” doctrine.

  • A null set is a set of measure zero (with respect to the measure generated by α).
  • A property holds almost everywhere (a.e.) if it fails only on a null set.

The key working rule:

If two functions are equal a.e., then they have the same integral (whenever either integral makes sense).

This is the engine behind many simplifications in analysis: you can freely redefine functions on null sets without changing integrals.

5.3 Convergence theorems

Convergence is the crown jewel of Lebesgue theory.

Three theorems form the practical backbone:

  • Monotone Convergence (MCT): increasing sequence of nonnegative functions ⇒ integral of the limit is the limit of the integrals.
  • Fatou’s Lemma: integral of liminf ≤ liminf of integrals.
  • Dominated Convergence (DCT): if fₙ → f a.e. and fₙ ≤ g with ∫g < ∞, then ∫fₙ → ∫f.

The book notes that full proofs are technically involved; what matters operationally is learning to recognize when the hypotheses are satisfied.

5.4 Extensions of the theory

Here the theory is pushed beyond the initial setup:

  • broader classes of functions,
  • extension from step/simple approximations to more general measurable behavior,
  • and the machinery you need to move toward functional analysis (next chapters).

Mental model

Riemann integration is good for geometry.

Lebesgue–Stieltjes integration is good for analysis, because it is designed to commute with the limiting operations you actually perform:

  • passing to limits of approximations,
  • exchanging limits and integrals under domination,
  • treating “pointwise disasters” as harmless if they live on null sets.

Practice checklist

  • Prove that changing a function on a countable set often doesn’t change the integral.
  • Practice DCT/MCT: identify the dominating function or the monotone structure.
  • Build comfort with “a.e.” reasoning: it is the native language of the subject.