1 Definition and basic idea
The Stieltjes integral extends ordinary integration by allowing accumulation to be measured relative to a function rather than directly with respect to interval length. In standard notation, it is written as \(\int f\,dg\), where \(f\) is the integrand and \(g\) is the integrator. This viewpoint makes it possible to describe sums, jumps, and other nonuniform changes within a single framework.
1.1 Informal interpretation
Ordinary integration can be viewed as adding up contributions over small pieces of the real line. The Stieltjes form replaces the uniform length increment with a change in \(g\). If \(g\) increases rapidly, the integral gives more weight to that region; if \(g\) is constant on an interval, that portion contributes nothing. This makes the construction especially natural for stepwise or irregular accumulation.
1.2 The role of the integrator function
The integrator \(g\) determines how the accumulation is measured. It may be smooth, monotone, or piecewise constant, and in many settings it need not be differentiable. When \(g\) has jumps, the integral can capture discrete contributions. When \(g\) varies continuously, the Stieltjes integral often behaves like a weighted ordinary integral.
1.3 Relation to the ordinary Riemann integral
The ordinary Riemann integral is recovered as a special case when \(g(x)=x\). In that situation, \(\int f\,dg\) becomes \(\int f(x)\,dx\). More generally, if \(g\) is differentiable with derivative \(g'\), then under suitable conditions the Stieltjes integral can be expressed as \(\int f(x)g'(x)\,dx\), showing that it includes the classical integral as a limiting form.
2 Riemann–Stieltjes integral
The Riemann–Stieltjes integral is defined through sums over partitions of an interval. It generalizes Riemann integration by replacing interval lengths with increments of the integrator function.
2.1 Partition sums
To approximate \(\int_a^b f\,dg\), one subdivides \([a,b]\) into subintervals and forms sums of the form \(\sum f(\xi_i)\bigl(g(x_i)-g(x_{i-1})\bigr)\). These sums estimate the total accumulation by sampling \(f\) at selected points and weighting by the change in \(g\) on each subinterval.
2.1.1 Tagged partitions
A tagged partition consists of points \(a=x_0<x_1<\cdots<x_n=b\) together with a chosen tag \(\xi_i\in[x_{i-1},x_i]\) for each subinterval. The corresponding Riemann–Stieltjes sum uses the tag value \(f(\xi_i)\) and the increment \(g(x_i)-g(x_{i-1})\). The choice of tags can affect intermediate sums, though the integral—when it exists—is independent of the choice in the limit.
2.1.2 Refinement of partitions
A refinement adds more subdivision points to a partition, producing a finer mesh. For Riemann–Stieltjes integration, finer partitions usually improve the approximation of the integral when the functions satisfy appropriate regularity conditions. Convergence under refinement is central to the existence of the integral.
2.2 Existence of the integral
The Riemann–Stieltjes integral does not exist for every pair of functions. Its existence depends on the behavior of both \(f\) and \(g\), especially near points of discontinuity or rapid variation.
2.2.1 Continuity conditions
A common sufficient condition is that \(f\) be continuous and \(g\) have bounded variation, or that \(g\) be continuous and \(f\) be of bounded variation. More refined results allow limited discontinuities, provided the discontinuities of the two functions do not interact too severely. In many practical cases, continuity of one function and monotonicity of the other is enough.
2.2.2 Bounded variation conditions
Bounded variation is a key criterion for existence and stability. If the integrator \(g\) has bounded variation on \([a,b]\), then the integral exists for every continuous \(f\). This condition ensures that the increments of \(g\) cannot oscillate too wildly, making the associated sums controllable.
2.3 Basic examples
Simple examples illustrate how the Stieltjes integral captures both continuous and discrete accumulation.
2.3.1 Step functions
If \(g\) is a step function, then \(\int f\,dg\) reduces to a finite sum over the jump points of \(g\). Each jump contributes the value of \(f\) near that point multiplied by the size of the jump. This case shows the direct connection between Stieltjes integration and weighted summation.
2.3.2 Piecewise smooth functions
When \(g\) is piecewise differentiable, the integral can often be computed by integrating \(f(x)g'(x)\) on each smooth segment and adding the contributions from jumps. This gives a practical method for many calculations and illustrates how the construction combines continuous and discrete effects.
3 Properties of the integral
The Riemann–Stieltjes integral retains many familiar properties of ordinary integration, though the precise statements depend on the regularity of the functions involved.
3.1 Linearity
The integral is linear in the integrand. If \(f_1\) and \(f_2\) are integrable with respect to \(g\), then so is any linear combination \(af_1+bf_2\), and \[ \int (af_1+bf_2)\,dg=a\int f_1\,dg+b\int f_2\,dg. \] This mirrors the corresponding property of standard integrals.
3.2 Additivity over intervals
If \(c\in[a,b]\), then the integral over \([a,b]\) can be split into integrals over \([a,c]\) and \([c,b]\), provided the relevant integrals exist. This additivity reflects the local nature of accumulation and is useful for computation on intervals with different behavior.
3.3 Integration by parts
A central identity is the Stieltjes form of integration by parts: \[ \int_a^b f\,dg+\int_a^b g\,df=f(b)g(b)-f(a)g(a), \] under suitable hypotheses. This formula generalizes the classical rule and remains valid in many cases where one or both functions are not differentiable. It is especially important in analysis and probability.
3.4 Comparison and estimation results
| Estimates often depend on the total variation of \(g\). If \(f\) is bounded and \(g\) has bounded variation, then the integral can be bounded in absolute value by the product of the supremum of \( | f | \) and the variation of \(g\). Such inequalities help establish convergence and control approximation errors. |
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4 Functions of bounded variation
Bounded variation provides the natural regularity class for many Stieltjes integrals. It measures how much a function can oscillate across an interval.
4.1 Definition and examples
| A function has bounded variation on \([a,b]\) if the supremum of the sums \(\sum | g(x_i)-g(x_{i-1}) | \) over all partitions is finite. Monotone functions are basic examples, as are piecewise monotone and piecewise smooth functions on compact intervals. Functions with infinitely many large oscillations typically fail to have bounded variation. |
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4.2 Decomposition into monotone parts
Any function of bounded variation can be written as the difference of two increasing functions. This representation is known as the Jordan decomposition. It is useful because monotone functions are easier to analyze and integrate, and it clarifies why bounded variation is the right setting for many Stieltjes results.
4.3 Connection with integrability
Functions of bounded variation interact well with Riemann–Stieltjes integration. If the integrator has bounded variation, then every continuous integrand is Stieltjes integrable. More generally, bounded variation helps ensure that the integral is stable under approximation and that the associated sums do not diverge due to oscillation.
5 Lebesgue–Stieltjes integral
The Lebesgue–Stieltjes integral reformulates Stieltjes integration using measure theory. It extends the idea from interval sums to integration with respect to a measure derived from an appropriate function.
5.1 Construction from a function of bounded variation
From a suitably regular monotone function, one can construct a measure on the real line by assigning mass to intervals according to the function’s increments. For a function of bounded variation, the construction is obtained by decomposing it into monotone parts and associating measures accordingly. This creates a powerful framework for integrating more general functions.
5.2 Associated measures
The measure arising from a Stieltjes function records how much the function increases over sets. When the function is nondecreasing, increments define a positive measure. The Lebesgue–Stieltjes integral then becomes an ordinary Lebesgue integral with respect to that measure, allowing the use of measure-theoretic tools such as convergence theorems.
5.3 Relationship with the Riemann–Stieltjes integral
When both integrals are defined, the Lebesgue–Stieltjes and Riemann–Stieltjes integrals often agree for sufficiently regular functions. The measure-theoretic version is more flexible and handles broader classes of integrands. The Riemann–Stieltjes integral, by contrast, is often easier to visualize through partition sums.
5.4 Absolutely continuous and singular parts
A Lebesgue–Stieltjes measure may decompose into parts corresponding to absolutely continuous variation, jump discontinuities, and singular continuous behavior. The absolutely continuous part has a density with respect to ordinary length, while the singular part concentrates on sets of zero Lebesgue measure. This decomposition helps classify how a function accumulates change.
6 Special cases and applications
Stieltjes integrals appear in settings where accumulation is discrete, irregular, or governed by distributions rather than smooth densities.
6.1 Summation by Stieltjes integral
Finite and infinite sums can often be expressed as Stieltjes integrals with stepwise integrators. This reformulation is useful for deriving summation formulas and comparing discrete and continuous accumulation. It also provides a bridge between counting and integration.
6.2 Distribution functions in probability
In probability theory, distribution functions define Stieltjes measures, and expectations can be written as Lebesgue–Stieltjes integrals. This is particularly useful for random variables with mixed discrete and continuous distributions. The formalism unifies point masses and density-based contributions.
6.3 Calculus with discontinuous data
The Stieltjes framework is well suited to models where input data change abruptly. It can represent cumulative quantities with jumps, such as inventories, signal levels, or piecewise constant processes. In such cases, ordinary differentiation may fail, but Stieltjes integration still gives a meaningful accumulation rule.
6.4 Spectral and analytical applications
In analysis, Stieltjes integrals appear in spectral representations, transform formulas, and the study of operators. They are used to encode measures arising from eigenvalue distributions and related cumulative data. The method is valuable because it naturally handles both continuous spectra and discrete contributions.
7 Generalizations
The Stieltjes idea extends beyond real-valued functions on intervals. It has analogues in abstract measure spaces, vector-valued settings, and higher-dimensional analysis.
7.1 Integration with respect to measures
Measure-theoretic integration generalizes the Stieltjes perspective by treating the integrator as a measure rather than a scalar function. In this form, the integral of a function depends on how the measure assigns weight to sets. The Lebesgue integral can be viewed as the broadest standard version of this idea.
7.2 Vector-valued Stieltjes integration
In some settings, the integrator or integrand may take values in a vector space. Vector-valued Stieltjes integration is used in functional analysis and in the theory of differential equations with irregular inputs. The main challenge is defining convergence and estimating variation in a way compatible with the geometry of the space.
7.3 Stieltjes integrals in higher dimensions
Higher-dimensional analogues replace intervals with regions in \(\mathbb{R}^n\) and use multidimensional measures or differential forms to express accumulation. These generalizations appear in geometry, analysis, and partial differential equations. Although the notation becomes more complex, the underlying principle remains the same: accumulation is measured relative to a chosen governing function or measure.