Integration theory rests on order, completeness, and limits. This chapter isolates those foundations because every later construction—from outer measure to $L^p$ convergence—uses them explicitly.

Scope and notation

Throughout the series, $\mathbb N={1,2,\ldots}$, $\mathbb Q$ denotes the rational numbers, and

\[\infty,\,-\infty\in\overline{\mathbb R} :=\mathbb R\cup\{-\infty,+\infty\}.\]

An interval written $(a,b]$ is open at $a$ and closed at $b$. This convention will later align with a right-continuous integrator $\alpha$ through $\mu_\alpha((a,b])=\alpha(b)-\alpha(a)$.

Completeness is the decisive property

The real numbers form a complete ordered field. The operative statement is the least-upper-bound axiom:

Every nonempty set $E\subset\mathbb R$ that is bounded above has a unique supremum $\sup E\in\mathbb R$.

The infimum follows by $\inf E=-\sup(-E)$. Completeness is stronger than the algebraic field laws; it is precisely what $\mathbb Q$ lacks. For example,

\[E=\{q\in\mathbb Q:q^2<2\}\]

is nonempty and bounded above in $\mathbb Q$, but has no rational supremum. In $\mathbb R$, its supremum is $\sqrt2$.

This axiom drives the monotone convergence principle. If $x_1\le x_2\le\cdots$ and $(x_n)$ is bounded above, set $x=\sup_n x_n$. For every $\varepsilon>0$, $x-\varepsilon$ cannot be an upper bound, so some $x_N>x-\varepsilon$; monotonicity then gives

\[x-\varepsilon<x_n\le x,\qquad n\ge N.\]

Hence $x_n\to x$. Measure theory repeatedly lifts this order argument from numbers to sets and functions.

Density and countability

The Archimedean property implies that for $a<b$ one can choose $n\in\mathbb N$ with $n(b-a)>1$, then an integer $m$ satisfying

\[na<m<nb.\]

Thus $a<m/n<b$, proving that $\mathbb Q$ is dense in $\mathbb R$. Because $\sqrt2\,\mathbb Q$ is also dense and contains irrationals away from zero, the irrationals are dense as well.

Density and cardinality must not be confused:

  • $\mathbb Q$ is countable and dense.
  • $\mathbb R\setminus\mathbb Q$ is uncountable and dense.
  • A set can therefore be topologically pervasive while remaining small for a particular measure.

The last qualification matters. Every countable set has Lebesgue measure zero, but a countable set need not have zero Lebesgue–Stieltjes measure. If

\[\alpha(x)=\sum_{k=1}^{\infty}2^{-k}\mathbf 1_{[k,\infty)}(x),\]

then $\mu_\alpha({k})=2^{-k}$ and $\mu_\alpha(\mathbb N)=1$. “Countable implies null” is valid for nonatomic measures such as Lebesgue measure, not for arbitrary Stieltjes measures.

Extended real values

The extended real line makes monotone limits and integrals of nonnegative functions total: they always exist in $[0,+\infty]$. Its order is natural,

\[-\infty < x < +\infty \quad (x\in\mathbb R),\]

but its arithmetic is deliberately partial. Expressions such as $+\infty-\infty$ and $0\cdot\infty$ are undefined. This is not a technical nuisance. It explains why the integral of a signed function is defined only when at least one of $\int f^+\,d\mu$ and $\int f^-\,d\mu$ is finite, and why finite integrability requires both to be finite.

For a sequence $(x_n)\subset\overline{\mathbb R}$,

\[\limsup_{n\to\infty}x_n =\inf_{N\ge1}\sup_{n\ge N}x_n,\qquad \liminf_{n\to\infty}x_n =\sup_{N\ge1}\inf_{n\ge N}x_n.\]

These values always exist in $\overline{\mathbb R}$, and $x_n$ converges exactly when the two agree.

Worked example: an infimum not attained

Consider $E=(0,1)$. Then $\inf E=0$ and $\sup E=1$, although neither endpoint belongs to $E$. An extremum is a member of the set; a supremum or infimum need only be the sharp order bound. The distinction is essential in optimization:

\[\inf_{x\in E}F(x)\]

can exist without a minimizer. Existence of a minimizing design therefore needs additional structure, commonly compactness of the admissible set and lower semicontinuity of $F$.

Computational interpretation

Exact completeness is a property of $\mathbb R$, not of floating-point numbers. A floating-point type is finite and discrete; rounding can destroy associativity and order comparisons near a tolerance. Reliable geometric computation therefore separates:

  • the mathematical existence statement, established in $\mathbb R$;
  • the approximation algorithm, which produces a finite representation;
  • the error contract, which states the metric, scale, and tolerance under which the result is accepted.

This separation becomes particularly important in CAD algorithms. Intersection points, closest-point parameters, and extremal curvature values may be defined through a supremum or infimum even when the numerical routine can only approximate them. The theorem establishes what the target is; numerical analysis establishes whether the implementation reaches it with a controlled error.

What this chapter establishes

The reusable foundation is now precise: completeness turns bounded monotone processes into limits; density supports approximation; countability does not by itself determine measure; and extended real values allow convergence theorems to be stated without artificial finite bounds. The next chapter transfers these ideas from numbers to functions.