The passage from real numbers to integration requires a controlled vocabulary for limits, discontinuities, variation, and approximation. These concepts describe which functions can generate Stieltjes measures and which algebraic operations preserve integrability.

Monotone sequences and series

For a nondecreasing sequence $(x_n)$,

\[x_n\longrightarrow \sup_{n\ge1}x_n\]

whenever the sequence is bounded above; otherwise $x_n\to+\infty$ in the extended real sense. This order form of convergence is the scalar prototype of the Monotone Convergence Theorem.

A double series $\sum_{m,n}a_{mn}$ needs a stated summation rule. If $a_{mn}\ge0$, all iterated and rectangular limits agree in $[0,+\infty]$:

\[\sum_m\sum_n a_{mn} =\sum_n\sum_m a_{mn}.\]

The same equality holds for signed terms when $\sum_{m,n}|a_{mn}|<\infty$. Without nonnegativity or absolute convergence, rearrangement can change the value or destroy convergence. This is the discrete form of the Tonelli–Fubini distinction developed later.

One-sided limits and monotone functions

If $\alpha:[a,b]\to\mathbb R$ is nondecreasing, then for every $x\in(a,b)$,

\[\alpha(x-)=\sup_{t<x}\alpha(t),\qquad \alpha(x+)=\inf_{t>x}\alpha(t)\]

exist and satisfy $\alpha(x-)\le\alpha(x)\le\alpha(x+)$. Every discontinuity is a jump.

Moreover, a monotone function has at most countably many discontinuities. To see the mechanism, fix $k\in\mathbb N$ and collect points whose jump exceeds $1/k$. On a bounded interval there can be only finitely many such points, because the sum of their jumps cannot exceed $\alpha(b)-\alpha(a)$. Taking the union over $k$ gives a countable set.

For the Stieltjes construction used in this series, the canonical integrator is nondecreasing and right-continuous. Its point mass is

\[\mu_\alpha(\{x\})=\alpha(x)-\alpha(x-).\]

Changing $\alpha$ by a constant changes no interval increment and therefore produces the same measure.

Step functions and measurable approximation

A step function on $[a,b]$ has the form

\[s(x)=\sum_{j=1}^{N}c_j\mathbf 1_{I_j}(x),\]

where the $I_j$ are intervals forming a finite partition. Step functions are computationally explicit, but simple measurable functions are the more general measure-theoretic objects:

\[\varphi(x)=\sum_{j=1}^{N}c_j\mathbf 1_{E_j}(x),\]

where each $E_j$ is measurable.

For every nonnegative measurable $f$, there is an increasing sequence of nonnegative simple functions $\varphi_n\uparrow f$. A standard construction quantizes both range and truncation:

\[\varphi_n(x) =2^{-n}\!\left\lfloor 2^n\min(f(x),n)\right\rfloor .\]

Then $0\le\varphi_n\le f$ and $\varphi_n(x)\to f(x)$ pointwise. This formula gives a concrete bridge between finite sums and the integral of an arbitrary nonnegative measurable function.

Positive and negative parts

Every real-valued function decomposes canonically as

\[f=f^+-f^-, \qquad |f|=f^++f^-,\]

with

\[f^+=\max(f,0),\qquad f^-=\max(-f,0).\]

Both parts are nonnegative and never simultaneously positive. The decomposition prevents the undefined expression $+\infty-\infty$. A measurable $f$ is integrable exactly when

\[\int |f|\,d\mu =\int f^+\,d\mu+\int f^-\,d\mu<\infty.\]

Bounded variation and Jordan decomposition

For $f:[a,b]\to\mathbb R$, define the total variation

\[V_a^b(f) =\sup_P\sum_{j=1}^{n}|f(x_j)-f(x_{j-1})|,\]

where the supremum is over all finite partitions $P:a=x_0<\cdots<x_n=b$.

The function has bounded variation when $V_a^b(f)<\infty$. Every monotone function has bounded variation, with $V_a^b(f)=|f(b)-f(a)|$. More substantially, the Jordan decomposition theorem states that $f$ has bounded variation if and only if it is the difference of two bounded nondecreasing functions.

One canonical construction uses the variation function $v(x)=V_a^x(f)$:

\[f(x) =f(a)+\frac{v(x)+f(x)-f(a)}{2} -\frac{v(x)-f(x)+f(a)}{2}.\]

The two nonconstant terms on the right are nondecreasing. This decomposition later extends Stieltjes integration from monotone integrators to signed measures generated by functions of bounded variation.

Absolute continuity

A function $f:[a,b]\to\mathbb R$ is absolutely continuous if for every $\varepsilon>0$ there is $\delta>0$ such that every finite disjoint family of intervals $(a_k,b_k)$ satisfying

\[\sum_k(b_k-a_k)<\delta\]

also satisfies

\[\sum_k|f(b_k)-f(a_k)|<\varepsilon.\]

Absolute continuity is strictly stronger than uniform continuity and bounded variation. Its structural characterization is fundamental:

$f$ is absolutely continuous on $[a,b]$ if and only if there exists $g\in L^1([a,b])$ such that \(f(x)=f(a)+\int_a^x g(t)\,dt.\) In that case $f’=g$ almost everywhere.

The Cantor function shows why the qualification matters: it is continuous, nondecreasing, and of bounded variation; its derivative is zero almost everywhere, yet it is not constant. Therefore bounded variation alone does not permit recovery from the almost-everywhere derivative.

Computational interpretation

These function classes correspond to distinct models of accumulated quantity:

  • an absolutely continuous component describes distributed density;
  • a jump component describes concentrated events or point loads;
  • a singular continuous component describes continuous accumulation supported on a Lebesgue-null set.

In engineering computation, treating all three as an ordinary density $w(x)\,dx$ loses information. A Stieltjes integrator preserves the distinction. For CAD and simulation, that distinction can represent distributed material, concentrated constraints, or parameter events without forcing them into one numerically misleading data model.