This book starts before integration, because Lebesgue-style arguments lean heavily on the structure of the real line and on limiting processes.

1) Dense sets: rationals and irrationals

Two facts that keep resurfacing:

  • Between any two real numbers a < b, there exist both a rational and an irrational number.
  • In fact, there are infinitely many of each between any two reals.

Why this matters: density is what makes “arbitrarily fine approximation” possible. Later, step functions approximate measurable functions by carving the domain into small intervals; density ensures those partitions behave as expected.

2) Countable vs. uncountable

The chapter reminds you of a hierarchy:

  • ℕ, ℤ, ℚ are countable (their elements can be listed as a sequence).
  • Any nontrivial real interval, e.g. (0, 1), is uncountable.

A practical takeaway you’ll use later: countable sets are “small” in measure theory. In Lebesgue integration, countable sets typically become null sets under reasonable measures, so changing a function on such a set doesn’t affect integrals.

3) The extended real line

It’s convenient to work with

  • ℝₑ = ℝ ∪ {+∞, −∞}

and allow intervals like (−∞, b] or (a, +∞).

This is not just notation. It keeps statements uniform when you later define measures and integrals on arbitrary intervals, not only closed bounded ones.

4) Supremum and infimum

A key completeness property is packaged as:

  • Every nonempty subset of ℝₑ has a least upper bound (supremum) and a greatest lower bound (infimum).

Two patterns to internalize:

  • sup S may or may not belong to S.
  • When sup S is finite, it can be characterized by the “ε-approximation” condition: for every ε > 0, there exists x ∈ S with sup S − ε < x ≤ sup S.

This “for every ε, exists x” template is the same logical shape you’ll see later in:

  • definitions of limits,
  • definitions of integrals via approximation,
  • convergence theorems.

Mental model for later chapters

If you want a one-line bridge to integration:

Integration is controlled approximation, and controlled approximation is powered by completeness + limits.

Lebesgue–Stieltjes theory will generalize “length” to a measure defined via a monotone function α. This only works smoothly when the underlying number system supports sup/inf, one-sided limits, and carefully defined intervals.

Practice checklist

Before moving on, make sure you can do these quickly:

  • Use density to find rationals/irrationals inside arbitrary intervals.
  • Prove basic countability facts (unions, images under simple maps).
  • Compute sup/inf for simple sets and verify the ε-characterization.