Riemann-Stieltjes integration replaces interval length by increments of an integrator. It is the deterministic ancestor of integration against measures and distribution functions.
Stieltjes sums
The integrator decides how much weight each subinterval receives.
Definition. For a tagged partition \[ P:a=x_0<\cdots<x_n=b,\qquad t_i\in[x_{i-1},x_i], \] the Riemann-Stieltjes sum is \[ S(f,\alpha;P,t)=\sum_{i=1}^n f(t_i)\bigl(\alpha(x_i)-\alpha(x_{i-1})\bigr). \] We say \(\int_a^b f\,\mathrm{d}\alpha=I\) if for every \(\varepsilon>0\) there is \(\delta>0\) such that \(\|P\|<\delta\) implies \(|S(f,\alpha;P,t)-I|<\varepsilon\) for every choice of tags.
Remark. When \(\alpha(x)=x\), this is the Riemann integral. When \(\alpha\) has jumps, the integral includes weighted point masses.
Monotone integrators and induced measures
Increasing integrators are distribution functions of finite measures on intervals.
Theorem. If \(\alpha\) is increasing and right-continuous, there is a unique finite Borel measure \(\mu_\alpha\) on \([a,b]\) such that \(\mu_\alpha((s,t])=\alpha(t)-\alpha(s)\).
Proof. After fixing the harmless normalization \(\alpha(a)=0\), define \[ \mu_0((s,t])=\alpha(t)-\alpha(s),\qquad \mu_0(\{a\})=0, \] and extend additively to the algebra of finite disjoint unions of such intervals. Right-continuity gives continuity from above, which is the key step in proving countable additivity of \(\mu_0\) on this algebra. The Carath?odory extension theorem produces a finite Borel measure \(\mu_\alpha\). Two finite measures agreeing on the generating \(\pi\)-system of half-open intervals agree on all Borel sets by the \(\pi\)-\(\lambda\) theorem.
Existence for continuous integrands
Continuous functions integrate against bounded-variation integrators.
Proposition. If \(f\) is continuous and \(\alpha\) has bounded variation on \([a,b]\), then \(\int f\,\mathrm{d}\alpha\) exists.
Proof. First suppose \(\alpha\) is increasing. Uniform continuity of \(f\) gives, for two choices of tags in the same partition, \[ |S(f,\alpha;P,t)-S(f,\alpha;P,s)| \le \omega_f(\|P\|)\bigl(\alpha(b)-\alpha(a)\bigr)\longrightarrow0. \] Step-function approximations then show that these sums converge to \(\int f\,\mathrm{d}\mu_\alpha\). For general bounded-variation \(\alpha\), its variation function \(v(x)=V_a^x(\alpha)\) is increasing and \[ \alpha=\frac{v+\alpha}{2}-\frac{v-\alpha}{2} \] is a difference of increasing functions. Define the integral by subtracting the two corresponding integrals; the same sum estimate proves consistency with the tagged definition.
Integration by parts
Stieltjes integration has a symmetric product rule.
Theorem. If \(f\) and \(g\) have bounded variation and share no common discontinuity, then \[ \int_a^b f\,\mathrm{d}g+\int_a^b g\,\mathrm{d}f=f(b)g(b)-f(a)g(a). \]
Proof. On each partition interval, \[ f(x_i)g(x_i)-f(x_{i-1})g(x_{i-1}) =f(x_{i-1})\Delta g_i+g(x_i)\Delta f_i. \] Summing telescopes. As the mesh tends to zero, the first sum converges to \(\int f\,\mathrm{d}g\) and the second to \(\int g\,\mathrm{d}f\); replacing right by left endpoint values creates only products of simultaneous jumps. The assumption of no common discontinuity makes their total limit zero, yielding the formula.
Remark (common jumps). With a fixed càdlàg convention, the general formula contains a correction: \[ \int_{(a,b]} f_-\,\mathrm{d}g+\int_{(a,b]}g\,\mathrm{d}f =f(b)g(b)-f(a)g(a)-\sum_{a<t\le b}\Delta f(t)\Delta g(t). \] The simple symmetric identity therefore requires either continuity of one integrator or an explicit jump convention.
Distribution functions
Stieltjes integration is the natural notation for expectations of functions of a real random variable.
Example. If \(F\) is a distribution function and \(X\sim F\), then \[ \mathbb E\varphi(X)=\int_{\mathbb R}\varphi\,\mathrm{d}F \] whenever either side is defined. At an atom \(x\), the jump \(\Delta F(x)=F(x)-F(x-)\) contributes \(\varphi(x)\Delta F(x)\). If \(F\) is absolutely continuous with density \(p\), the same integral becomes \(\int\varphi(x)p(x)\,\mathrm{d}x\); discrete, continuous, and mixed laws are thus handled by one notation.