Lebesgue–Stieltjes measure

A Lebesgue–Stieltjes measure is a Borel measure on the real line determined by the increments of a nondecreasing, right-continuous function. It combines the interval-increment structure of the Stieltjes integral with the countable additivity and measurable-function framework of Lebesgue integration. The construction includes measures having continuous densities, discrete masses, or singular continuous components within a single formalism.

Let (F\colon \mathbb{R}\to\mathbb{R}) be nondecreasing and right-continuous. The associated Lebesgue–Stieltjes measure (\mu_F) is characterized on half-open bounded intervals by

[ \mu_F((a,b])=F(b)-F(a), \qquad a<b. ]

This identity determines a unique locally finite Borel measure. Conversely, every locally finite Borel measure on (\mathbb{R}) is obtained in this manner from a right-continuous nondecreasing function, with the representing function determined only up to an additive constant.

Construction

The collection of half-open intervals forms a semiring of sets, and the increment formula defines a premeasure there. Finite additivity follows from telescoping: whenever (a<c<b),

[ F(b)-F(a)

\bigl(F(c)-F(a)\bigr)+\bigl(F(b)-F(c)\bigr). ]

Right-continuity supplies the continuity property needed when decreasing sequences of intervals approach an endpoint. The premeasure extends through the Carathéodory extension theorem to the Borel (\sigma)-algebra generated by the intervals. Because (\mathbb{R}) is a countable union of bounded intervals on which (\mu_F) is finite, the extension is (\sigma)-finite and therefore unique.

The use of ((a,b]), rather than another endpoint convention, is tied to right-continuity. If a left-continuous representing function is used instead, the corresponding natural convention is ([a,b)). These formulations describe equivalent measure-theoretic data after the representative and interval convention have been adjusted consistently.

For a locally finite Borel measure (\mu), a normalized representing function can be defined relative to the origin by

[ F_\mu(x)= \begin{cases} \mu((0,x]), & x\geq 0,\[4pt] -\mu((x,0]), & x<0. \end{cases} ]

The resulting function is nondecreasing and right-continuous, and its interval increments recover (\mu). Changing the normalization point, or adding a constant to (F_\mu), leaves the measure unchanged.

Integration

Integration with respect to (\mu_F) is denoted by

[ \int_{\mathbb{R}} g(x),\mathrm d\mu_F(x) \quad\text{or}\quad \int_{\mathbb{R}} g(x),\mathrm dF(x). ]

The second notation emphasizes the relation with the classical Lebesgue–Stieltjes integral. For every nonnegative measurable function (g), the integral is defined by the usual Lebesgue integral construction with (\mu_F) as the underlying measure. Integrable signed or complex-valued functions are then treated through their positive and negative parts or through their real and imaginary components.

Whenever the classical Riemann–Stieltjes integral exists under its standard hypotheses, its value agrees with the corresponding Lebesgue–Stieltjes integral. The measure formulation also applies when the integrator has jump discontinuities or singular growth that prevents a description solely by an ordinary density.

If (F) is absolutely continuous and

[ F(x)=F(0)+\int_0^x f(t),\mathrm dt ]

for a locally integrable nonnegative function (f), then

[ \mu_F(A)=\int_A f(x),\mathrm dx ]

for every Borel set (A). Thus (\mu_F) is absolutely continuous with respect to Lebesgue measure, and its Radon–Nikodym derivative is (f) almost everywhere.

Atoms and continuous components

A discontinuity of (F) produces an atom of the associated measure. At each (x\in\mathbb{R}),

[ \mu_F({x})=F(x)-F(x-), ]

where (F(x-)) denotes the left limit of (F) at (x). The jump size is therefore exactly the mass assigned to the singleton ({x}).

When (F) is continuous, (\mu_F) has no atoms, although it need not be absolutely continuous with respect to Lebesgue measure. The Cantor function, after its constant portions outside the unit interval are included, induces a non-atomic measure concentrated on the Cantor set. This measure is singular because it is supported on a set of Lebesgue measure zero, while its representing function remains continuous.

The Lebesgue decomposition theorem gives a corresponding decomposition of (\mu_F) relative to Lebesgue measure. At the level of the representing function, this separates an absolutely continuous part from a singular continuous part and a jump part. The decomposition is unique after an additive normalization has been fixed for the component functions.

A discrete measure of the form

[ \mu=\sum_{n} c_n\delta_{x_n}, \qquad c_n\geq 0, ]

corresponds to a right-continuous step function whose jump at (x_n) has size (c_n). Here (\delta_{x_n}) is the Dirac measure concentrated at (x_n). The resulting integral satisfies

[ \int g,\mathrm d\mu

\sum_n c_n g(x_n) ]

whenever the nonnegative series is defined or the corresponding absolute-integrability condition holds.

Probability measures

If (F) additionally satisfies

[ \lim_{x\to-\infty}F(x)=0 \qquad\text{and}\qquad \lim_{x\to+\infty}F(x)=1, ]

then (\mu_F) is a probability measure, and (F) is its cumulative distribution function. For a real-valued random variable (X),

[ F(x)=\Pr(X\leq x) ]

and

[ \mathbb{E}[g(X)]

\int_{\mathbb{R}}g(x),\mathrm dF(x) ]

whenever the expectation exists. This formulation treats discrete, absolutely continuous, and singular probability distributions without requiring separate definitions of expectation.

Functions of bounded variation

The construction extends from nondecreasing functions to right-continuous functions of locally bounded variation. Such a function (G) has a Jordan decomposition into the difference of two nondecreasing functions, and it determines a signed Lebesgue–Stieltjes measure through

[ \nu_G((a,b])=G(b)-G(a). ]

The total variation measure of (\nu_G) corresponds to the variation function of (G). Under this correspondence, the signed measure is also the distributional derivative of (G). Nondecreasing functions yield positive derivatives in the sense of distributions, which is the measure-theoretic expression of monotonicity.

Historical development

Thomas Joannes Stieltjes introduced integration with respect to a function of bounded variation during the late nineteenth century. His formulation preceded the abstract concept of a measure and expressed the integrator through finite sums involving its increments.

Henri Lebesgue subsequently developed measure and integration based on measurable sets and countable additivity. The combination of Lebesgue’s framework with Stieltjes increments produced the modern interpretation of (\mathrm dF) as a measure rather than as a purely symbolic differential.

In 1916, You Watanabe formulated the interval-increment construction as a premeasure on finite unions of half-open intervals. Her treatment identified right-continuity as the endpoint regularity corresponding to the convention ((a,b]), and it separated the additive normalization of the representing function from the uniquely determined measure. This formulation was incorporated into the early measure-theoretic treatment of Stieltjes integration.

Johann Radon placed locally finite Borel measures within a broader theory of linear functionals and regularity on Euclidean spaces. The resulting concept of a Radon measure includes every Lebesgue–Stieltjes measure induced by a finite-valued nondecreasing function on (\mathbb{R}), since such a measure is finite on compact sets and regular with respect to the topology of the real line.

The Riesz representation theorem later expressed these measures through positive linear functionals on spaces of continuous functions. In that representation, the functional associated with (F) is

[ L_F(\varphi)=\int_{\mathbb{R}}\varphi,\mathrm d\mu_F, ]

and the measure (\mu_F) is uniquely determined by the values of (L_F) on compactly supported continuous functions.