1 Background and motivation
The Lebesgue–Stieltjes integral is a framework for integrating a function against a monotone function, or more generally against the measure generated by such a function. It unifies several earlier notions of integration and is especially useful on the real line, where distribution functions, cumulative quantities, and functions of bounded variation arise naturally.
1.1 From Riemann–Stieltjes to Lebesgue–Stieltjes integration
The classical Riemann–Stieltjes integral assigns meaning to expressions of the form \(\int f\,dg\), where \(g\) is an integrator of bounded variation. Its definition is based on tagged partitions and works well for many elementary functions, but it can be limited when one wants stronger convergence theorems or more flexible measurability conditions. The Lebesgue–Stieltjes integral replaces partition sums with measure-theoretic integration, allowing the integrator to be treated as a measure rather than only as a function.
1.2 Functions of bounded variation
A function on an interval has bounded variation if its total oscillation, measured through partition sums, remains finite. Such functions may be decomposed into increasing parts, and they serve as natural integrators in both Riemann–Stieltjes and Lebesgue–Stieltjes settings. Bounded variation is important because it captures many functions used in analysis, including monotone functions, piecewise smooth functions, and distribution functions.
1.3 Measures induced by monotone functions
Any nondecreasing function on the real line determines a measure by assigning to each interval the increment of the function across that interval. This induced measure reflects the “mass” accumulated by the integrator. In probability theory, this construction turns a cumulative distribution function into a probability measure, making the integral a tool for computing averages and probabilities.
2 Construction of the integral
The Lebesgue–Stieltjes integral is constructed by first associating a measure to a monotone function and then applying standard measure-theoretic integration with respect to that measure. This approach makes the theory consistent with the general Lebesgue integral while retaining the special structure of one-dimensional distribution functions.
2.1 Stieltjes measures on the real line
Given a nondecreasing right-continuous function on \(\mathbb{R}\), one can define a measure on Borel sets so that intervals receive measure equal to the function’s increment over that interval. This measure is often called a Stieltjes measure. Right continuity ensures that the measure behaves correctly on limiting intervals and that the correspondence with the generating function is well behaved.
2.2 Integration with respect to a measure
Once the Stieltjes measure is available, the integral of a measurable function is defined in the usual measure-theoretic way. For nonnegative functions, the integral is built from approximations by simple functions. For general functions, one separates positive and negative parts and integrates each when possible. Thus the Lebesgue–Stieltjes integral is not a separate theory so much as an instance of Lebesgue integration with a special measure.
2.3 Definition for simple functions
A simple function takes only finitely many values and can be written as a finite sum of constants times indicator functions. Its integral with respect to a Stieltjes measure is computed by summing the constants weighted by the measure of the corresponding sets. This case provides the starting point for the general definition and makes the construction concrete.
2.4 Extension to measurable functions
Nonnegative measurable functions are handled by approximating them from below with increasing sequences of simple functions. The integral is defined as the supremum of the integrals of these approximants. For arbitrary measurable functions, the same procedure is applied separately to the positive and negative parts, with finiteness determining whether the integral exists.
3 Integrable functions
Not every measurable function is integrable with respect to a given Stieltjes measure. The conditions for integrability depend on how the function behaves relative to the measure, just as in ordinary Lebesgue theory.
3.1 Measurability requirements
To be integrable, a function must be measurable with respect to the underlying \(\sigma\)-algebra, usually the Borel or Lebesgue \(\sigma\)-algebra on the real line. Measurability ensures that the function can be approximated by simple functions and that the integral is defined through measurable level sets.
3.2 Positive and negative parts
Any real-valued measurable function \(f\) can be decomposed as \(f=f^+-f^-\), where \(f^+=\max(f,0)\) and \(f^-=\max(-f,0)\). The integral is defined when at least one of these parts is finite and the difference is meaningful. This decomposition is central for handling functions that take both signs.
3.3 Absolute integrability
A function is absolutely integrable if the integral of its absolute value is finite. Absolute integrability is a strong condition that guarantees good convergence properties and prevents cancellation from hiding large positive and negative contributions. Many standard theorems are simplest under this assumption.
3.4 Spaces of integrable functions
The set of integrable functions forms a linear space under mild hypotheses on the measure. When equipped with norms or seminorms based on absolute integrability, these spaces become central objects in analysis. They provide the natural context for studying convergence, approximation, and duality.
4 Fundamental properties
The Lebesgue–Stieltjes integral inherits the basic structural theorems of Lebesgue integration. These properties make it especially powerful for limit processes and for calculations involving interval decompositions.
4.1 Linearity
The integral is linear wherever it is defined: the integral of a sum is the sum of the integrals, and constants factor out. Linearity is one of the main reasons the integral is so useful in analysis, since it allows complicated expressions to be broken into simpler components.
4.2 Monotone convergence
If a sequence of nonnegative measurable functions increases pointwise to a limit, then the integrals of the sequence increase to the integral of the limit. This theorem permits the passage from simple approximations to more general functions and is fundamental to the construction of the integral.
4.3 Dominated convergence
If a sequence of measurable functions converges pointwise and is dominated in absolute value by an integrable function, then the integrals converge to the integral of the limit. This result is a key tool for interchanging limits and integrals in many arguments.
4.4 Additivity over intervals
When the integrator is associated with a measure on the real line, the integral over a set can often be decomposed into integrals over disjoint pieces. For intervals, this reflects the additivity of the underlying measure and the cumulative nature of the generating function.
4.5 Continuity from above and below
Measures are continuous under increasing unions and decreasing intersections under appropriate finiteness assumptions. These continuity properties transfer to the Lebesgue–Stieltjes integral and are frequently used to evaluate limits of interval-based expressions.
5 Relationship to other integrals
The Lebesgue–Stieltjes integral sits between classical Riemann-type integration and the general Lebesgue integral. It can reproduce familiar formulas while extending them to a broader setting.
5.1 Comparison with the Riemann–Stieltjes integral
When the integrand and integrator are sufficiently regular, the Lebesgue–Stieltjes and Riemann–Stieltjes integrals agree. The Lebesgue version is more flexible because it is defined through measure theory and therefore supports powerful convergence theorems that are not always available in the Riemann setting.
5.2 Comparison with the Lebesgue integral
The Lebesgue–Stieltjes integral is a Lebesgue integral with respect to a particular measure. In this sense it is not separate from Lebesgue integration, but rather a specialization that emphasizes one-dimensional cumulative structure. This viewpoint clarifies why its properties mirror those of the standard Lebesgue integral.
5.3 Connection with the classical Riemann integral
When the Stieltjes integrator is the identity function, the Lebesgue–Stieltjes integral reduces to the ordinary Lebesgue integral with respect to length. For continuous functions on compact intervals, this in turn agrees with the classical Riemann integral. Thus the theory includes the standard integral as a special case.
5.4 Integration by parts
An integration-by-parts formula relates \(\int f\,dg\) and \(\int g\,df\) under suitable hypotheses. Such formulas are especially useful when one of the functions has bounded variation or when one wants to transfer derivatives or differences from one factor to another. Boundary terms appear in the same manner as in classical calculus.
6 Distribution functions and probability
One of the most important uses of the Lebesgue–Stieltjes integral is in probability theory, where distribution functions naturally generate probability measures. This gives a concise way to express expectations and probabilities of events.
6.1 Cumulative distribution functions
A cumulative distribution function is nondecreasing, right-continuous, and ranges from 0 to 1 at the extremes. It determines a probability measure on the real line through its increments. The associated Lebesgue–Stieltjes integral provides a compact formulation of probabilistic quantities.
6.2 Expected value as a Lebesgue–Stieltjes integral
If a random variable has distribution function \(F\), then its expected value can be written as an integral with respect to the measure induced by \(F\). This representation is often more convenient than working directly with densities, especially when no density exists. It also applies to functions of random variables.
6.3 Discrete, continuous, and mixed distributions
The framework handles discrete distributions by assigning positive mass to isolated points, continuous distributions by spreading mass over intervals, and mixed distributions by combining both behaviors. In each case, the same integral formula applies. This uniformity is one of the major advantages of the Lebesgue–Stieltjes approach.
6.4 Random variables and induced measures
A random variable induces a measure on the real line through its distribution. Integration with respect to this induced measure describes averages of functions of the random variable. This viewpoint is central in modern probability, where distributions are treated as measures rather than merely as formulas.
7 Advanced topics
Beyond the basic theory, Lebesgue–Stieltjes integration connects with signed measures, decomposition theorems, and change-of-variable formulas. These refinements are essential in deeper parts of analysis.
7.1 Signed Stieltjes measures
When the integrator has bounded variation rather than being monotone, it can generate a signed measure instead of a positive measure. Signed measures allow positive and negative contributions and extend the integral to a broader class of integrators. Care is needed to ensure that the resulting integral is well defined.
7.2 Functions of bounded variation and Jordan decomposition
Every function of bounded variation can be written as the difference of two increasing functions. This is known as Jordan decomposition. It reduces many questions about general bounded-variation integrators to the monotone case and clarifies the structure behind signed Stieltjes measures.
7.3 Change of variables
A substitution formula relates integrals after a transformation of the variable, provided the transformation has suitable monotonicity or regularity properties. Such formulas are useful for translating integrals into more convenient coordinates. In one dimension, they often arise from composing distribution functions with monotone maps.
7.4 Absolute continuity and Radon–Nikodym connections
If a Stieltjes measure is absolutely continuous with respect to Lebesgue measure, then it has a density that can be integrated in the ordinary sense. The Radon–Nikodym theorem explains this relationship and identifies derivatives of cumulative functions almost everywhere. This connection links Lebesgue–Stieltjes integration to differentiation and density theory.
8 Applications
The Lebesgue–Stieltjes integral appears throughout analysis and its applications. Its strength lies in treating sums, densities, jumps, and singular behavior within one formalism.
8.1 Real analysis
In real analysis, the integral is used to study monotone functions, bounded variation, convergence theorems, and the fine structure of measures on the line. It provides a precise language for interval accumulation and for integrating against non-smooth data.
8.2 Probability theory
Probability theory uses the Lebesgue–Stieltjes integral to express expectations, distributional identities, and tail probabilities. It is especially valuable when random variables have atoms or singular components. The measure-theoretic formulation also underlies modern limit theorems.
8.3 Stochastic processes
For stochastic processes, Stieltjes-type integrals are closely related to cumulative paths, variation estimates, and pathwise integration against finite-variation processes. They also provide a foundation for more advanced stochastic calculus constructions, where measure and variation play distinct roles.
8.4 Spectral theory
In spectral theory, cumulative spectral functions and spectral measures are often represented in Stieltjes form. Integrals with respect to such measures describe functional calculus for operators and encode the distribution of spectral values. This makes the framework useful in functional analysis and mathematical physics.
8.5 Engineering and applied mathematics
In applications, the integral is used to model accumulated quantities, piecewise-constant signals, and systems with jumps. It appears in reliability theory, queueing models, and signal processing whenever data are naturally encoded by cumulative functions. The formalism is well suited to both deterministic and probabilistic models.
9 Examples
Concrete examples illustrate how the theory captures discrete, continuous, and singular behavior using a single integral concept.
9.1 Step functions
If the integrator is a step function, the induced measure is concentrated on finitely or countably many points. The integral then becomes a weighted sum of the values of the integrand at those points, making the connection with discrete distributions explicit.
9.2 Continuous increasing functions
When the integrator is continuous and increasing, the measure may be spread continuously over intervals. If the function is differentiable with integrable derivative, the Stieltjes integral often reduces to an ordinary integral against that derivative. This case generalizes classical weighted integration.
9.3 Singular measures
Some increasing functions generate measures that are continuous but concentrated on sets of Lebesgue measure zero. Such singular measures are important in illustrating that the Lebesgue–Stieltjes integral is strictly broader than ordinary integration with respect to length. They show that cumulative functions can encode highly nonclassical distributions.
9.4 Piecewise linear integrators
For a piecewise linear monotone integrator, the measure splits into intervals where the slope is constant and corners where the function changes direction. The resulting integral combines ordinary weighted integration on each linear segment with discrete contributions at jump points if present. This makes the example useful for computations and for intuition about mixed behavior.
</INTERNAL_LINK_CANDIDATES> Measure theory (the branch of mathematics defining measures and measurable sets) Lebesgue integral (the general integral with respect to a measure) Riemann–Stieltjes integral (an integral with respect to a bounded-variation function) Function of bounded variation (a function with finite total variation) Monotone function (a nondecreasing or nonincreasing function) Stieltjes measure (the measure induced by a monotone function) Simple function (a finite-valued measurable function used in integral construction) Measurable function (a function compatible with a measurable structure) Positive and negative parts (the decomposition of a real-valued function into nonnegative components) Absolute integrability (integrability of the absolute value of a function) Monotone convergence theorem (the theorem on increasing limits of nonnegative functions) Dominated convergence theorem (the theorem controlling limits under an integrable bound) Cumulative distribution function (the probability distribution’s accumulating function) Expected value (the probabilistic mean expressed as an integral) Signed measure (a measure allowing positive and negative values) Jordan decomposition (the representation of a bounded-variation function as a difference of increasing functions) Radon–Nikodym theorem (the theorem relating absolutely continuous measures to densities) Singular measure (a measure concentrated on a set of Lebesgue measure zero) Spectral theory (the study of operators via spectral measures) Integration by parts (a formula relating integrals of products and derivatives)