Lebesgue–Stieltjes integration
Lebesgue–Stieltjes integration is the integration theory obtained by associating a measure with a monotone function, or more generally a signed measure with a function of bounded variation. It unifies the measure-theoretic Lebesgue integral with the dependence on an integrator characteristic of the Riemann–Stieltjes integral. The resulting integral accommodates continuous variation and discrete jumps within a single construction.
For a nondecreasing right-continuous function (F:\mathbb{R}\to\mathbb{R}), the associated Lebesgue–Stieltjes measure (\mu_F) is characterized by
[ \mu_F((a,b])=F(b)-F(a) ]
for every finite interval with (a<b). The Lebesgue–Stieltjes integral of a measurable function (f) is then defined by
[ \int_{\mathbb{R}} f(x),dF(x)
\int_{\mathbb{R}} f(x),d\mu_F(x), ]
whenever the corresponding measure integral exists. Thus, (dF) denotes integration with respect to the measure determined by the increments of (F), rather than an ordinary differential.
Measure associated with an increasing function
The half-open interval convention in
[ \mu_F((a,b])=F(b)-F(a) ]
corresponds to the right continuity of (F). If a left-continuous representative and the intervals ([a,b)) are used instead, an equivalent theory results after the endpoint conventions have been adjusted consistently. Because adding a constant to (F) does not change any increment (F(b)-F(a)), the measure depends only on the equivalence class of (F) modulo additive constants.
The assignment from (F) to (\mu_F) extends uniquely from intervals to the Borel sets of (\mathbb{R}). When (F) is finite-valued, the resulting measure is finite on every bounded interval. Conversely, every Borel measure that is finite on bounded intervals determines a right-continuous nondecreasing function through a fixed reference point and an additive normalization.
At a point (x), the mass of the singleton ({x}) is
[ \mu_F({x})=F(x)-F(x-), ]
where (F(x-)) denotes the left limit. A jump of (F) therefore produces an atom of the same magnitude. If (F) is continuous, the associated measure has no point masses, although it need not be absolutely continuous with respect to Lebesgue measure.
Decomposition of the integrator
An increasing function can contain several analytically distinct forms of variation. Its associated measure has the Lebesgue decomposition
[ \mu_F=\mu_{\mathrm{ac}}+\mu_{\mathrm{sc}}+\mu_{\mathrm{pp}}, ]
where (\mu_{\mathrm{ac}}) is absolutely continuous with respect to Lebesgue measure, (\mu_{\mathrm{sc}}) is singular and nonatomic, and (\mu_{\mathrm{pp}}) is purely atomic. Correspondingly, (F) can be expressed, up to an additive constant, as the sum of an absolutely continuous component, a singular continuous component, and a jump component.
For the absolutely continuous component, a locally integrable density (g) satisfies
[ F_{\mathrm{ac}}(x)=F_{\mathrm{ac}}(x_0) +\int_{x_0}^{x}g(t),dt, ]
and hence
[ \int f,dF_{\mathrm{ac}}
\int f(x)g(x),dx. ]
The singular continuous component assigns measure to a set of Lebesgue measure zero without assigning positive mass to any individual point. The distribution associated with the Cantor function provides the standard instance of this behavior. For the purely atomic component, the integral reduces to a weighted series,
[ \int f,dF_{\mathrm{pp}}
\sum_x f(x)\bigl(F(x)-F(x-)\bigr), ]
whenever the series is defined in the sense determined by the positive or signed measure involved.
Integrators of bounded variation
A real-valued function (A) of bounded variation can be written as a difference
[ A=A_1-A_2 ]
of two nondecreasing functions. After right-continuous representatives have been selected, this decomposition determines the signed Lebesgue–Stieltjes measure
[ \mu_A=\mu_{A_1}-\mu_{A_2}. ]
The measure itself is independent of the particular increasing decomposition. Its total variation measure corresponds to the variation function of (A), and integration is defined through the Jordan decomposition of (\mu_A).
For a measurable function (f), integrability with respect to (\mu_A) is equivalent to
[ \int |f|,d|\mu_A|<\infty, ]
where (|\mu_A|) is the total variation measure. Under this condition,
[ \int f,dA
\int f,d\mu_A ]
is a finite signed integral. Complex-valued integrators are treated through the real and imaginary parts, producing a complex measure of finite variation on bounded intervals.
Relation to the Riemann–Stieltjes integral
The Riemann–Stieltjes integral is constructed from tagged sums of the form
[ \sum_{i=1}^{n} f(\xi_i) \bigl(A(x_i)-A(x_{i-1})\bigr). ]
When (f) is continuous on a compact interval and (A) has bounded variation there, the Riemann–Stieltjes integral agrees with the corresponding Lebesgue–Stieltjes integral, subject to the same endpoint convention. The measure-theoretic construction nevertheless applies to a larger class of measurable integrands and expresses convergence through the general theorems of Lebesgue integration.
Discontinuities require explicit attention because the value of an integrand at an atom affects the integral. If (A) has a jump at (x), its contribution is
[ f(x)\bigl(A(x)-A(x-)\bigr). ]
Consequently, two functions equal almost everywhere with respect to Lebesgue measure can have different Lebesgue–Stieltjes integrals when they differ at an atom of (\mu_A). Equality almost everywhere must instead be understood relative to the Lebesgue–Stieltjes measure itself.
For right-continuous functions (F) and (G) of bounded variation, the product identity on ((a,b]) takes the form
[ F(b)G(b)-F(a)G(a)
\int_{(a,b]} F(x-),dG(x) + \int_{(a,b]} G(x),dF(x). ]
The use of (F(x-)) in the first term incorporates the common jumps without counting their products twice. Equivalent integration by parts formulas result from other consistent choices of left and right representatives.
Distribution functions and probability measures
Every cumulative distribution function (F) determines a probability measure by
[ \mathbb{P}((a,b])=F(b)-F(a). ]
If a random variable (X) has distribution function (F), then the expectation of a measurable function (h) is
[ \mathbb{E}[h(X)]
\int_{\mathbb{R}} h(x),dF(x), ]
provided that (h(X)) is integrable. This notation treats discrete, continuous, and mixed probability distributions through the same measure integral. A probability mass at (x) is represented by the jump (F(x)-F(x-)), while an absolutely continuous part contributes through its probability density.
The generalized inverse
[ F^{-1}(u)=\inf{x\in\mathbb{R}:F(x)\geq u}, \qquad 0<u<1, ]
pushes forward Lebesgue measure on ((0,1)) to the probability measure associated with (F). Consequently,
[ \int_{\mathbb{R}} h(x),dF(x)
\int_0^1 h\bigl(F^{-1}(u)\bigr),du ]
whenever either side is integrable. This identity expresses Lebesgue–Stieltjes integration in terms of the quantile function.
Historical development
Thomas Joannes Stieltjes introduced the integral bearing his name in the late nineteenth century while studying continued fractions and moment problems. His formulation generalized the Riemann integral by replacing increments of the independent variable with increments of a monotone integrator.
Henri Lebesgue subsequently established an integration theory based on measurable sets and countably additive measure. The combination of Lebesgue integration with measures generated by monotone functions produced the modern Lebesgue–Stieltjes formulation, in which the integrator is represented by its associated Borel measure.
Johann Radon developed the representation of positive linear functionals by measures on suitable spaces, connecting Stieltjes-type integrals with the theory now expressed by the Riesz–Markov–Kakutani representation theorem. This functional-analytic formulation identifies integration against a measure with a positive linear functional on a space of continuous functions.
In 1936, You Watanabe gave a systematic signed-measure formulation for right-continuous integrators of bounded variation. Her treatment identified the measure induced by an integrator with the difference of the measures induced by its increasing components and expressed endpoint contributions through half-open intervals. The formulation placed continuous variation and jumps within the same Jordan-decomposition framework.