The Lebesgue–Stieltjes construction turns a nondecreasing function into a measure. Ordinary length, discrete point masses, and mixed distributions then become instances of one integration theory.

From an integrator to a measure

Let $\alpha:\mathbb R\to\mathbb R$ be nondecreasing and right-continuous. Define interval increments on half-open intervals by

\[\mu_\alpha((a,b])=\alpha(b)-\alpha(a).\]

Half-open intervals concatenate without overlap:

\[(a,c]=(a,b]\mathbin{\dot\cup}(b,c].\]

Consequently, the increment rule is finitely additive on the algebra generated by such intervals. The measure extension theorem supplies a unique Borel measure $\mu_\alpha$ agreeing with these increments, subject to the usual local finiteness inherited from finite-valued $\alpha$.

The right-continuity convention fixes the correspondence uniquely. If an initial monotone function is not right-continuous, one uses its right-continuous representative. Adding a constant to $\alpha$ leaves every increment unchanged and therefore leaves $\mu_\alpha$ unchanged.

The notation

\[\int f\,d\alpha\]

means $\int f\,d\mu_\alpha$.

Three canonical integrators

Continuous density

If

\[\alpha(x)=\int_{-\infty}^{x}w(t)\,dt,\qquad w\ge0,\]

then $\alpha$ is absolutely continuous and

\[d\mu_\alpha(x)=w(x)\,dx,\qquad \int f\,d\alpha=\int f(x)w(x)\,dx.\]

Lebesgue measure is the special case $\alpha(x)=x$, up to an additive constant.

One atom

For $c\in\mathbb R$, let $\alpha(x)=\mathbf 1_{[c,\infty)}(x)$. Then

\[\mu_\alpha=\delta_c,\qquad \int f\,d\alpha=f(c)\]

for every measurable $f$ for which the value is defined.

Mixed accumulation

Let

\[\alpha(x)=\int_{-\infty}^{x}w(t)\,dt +\sum_{k=1}^{\infty}a_k\mathbf 1_{[x_k,\infty)}(x), \qquad a_k\ge0,\quad \sum_k a_k<\infty.\]

Then

\[\int f\,d\alpha =\int f(x)w(x)\,dx+\sum_{k=1}^{\infty}a_k f(x_k)\]

whenever the nonnegative integral is defined, or when the signed expression is absolutely integrable. One formalism therefore represents distributed and concentrated contributions without encoding atoms as narrow artificial peaks.

The integral: simple functions first

For a nonnegative simple measurable function

\[\varphi=\sum_{j=1}^{N}c_j\mathbf 1_{E_j}, \qquad c_j\ge0,\]

with disjoint measurable sets $E_j$, define

\[\int \varphi\,d\mu_\alpha =\sum_{j=1}^{N}c_j\,\mu_\alpha(E_j).\]

The value is independent of the chosen representation. For a nonnegative measurable function $f$,

\[\int f\,d\mu_\alpha =\sup\left\{ \int\varphi\,d\mu_\alpha: 0\le\varphi\le f,\quad \varphi\text{ simple} \right\}.\]

This definition may equal $+\infty$. For a real measurable $f$, write $f=f^+-f^-$. The integral

\[\int f\,d\mu_\alpha =\int f^+\,d\mu_\alpha-\int f^-\,d\mu_\alpha\]

is defined when the right side does not have the indeterminate form $+\infty-\infty$. The function is integrable when

\[\int |f|\,d\mu_\alpha<\infty.\]

Probability distributions

If $F:\mathbb R\to[0,1]$ is nondecreasing, right-continuous, and satisfies

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

then $\mu_F$ is a probability measure and $F$ is its cumulative distribution function:

\[F(x)=\mu_F((-\infty,x]).\]

For a random variable $X$ having distribution $\mu_F$,

\[\mathbb E[g(X)]=\int_{\mathbb R}g(x)\,dF(x)\]

whenever $g$ is nonnegative or integrable. This formula remains valid for continuous, discrete, singular, and mixed distributions.

Worked example: a hybrid engineering quantity

On $[0,1]$, define

\[\alpha(x)=2x+3\mathbf 1_{[1/2,\,1]}(x).\]

The associated measure is

\[\mu_\alpha=2\lambda+3\delta_{1/2},\]

where $\lambda$ is Lebesgue measure. For $f(x)=x^2$,

\[\int_{(0,1]}x^2\,d\alpha(x) =2\int_0^1x^2\,dx+3\left(\frac12\right)^2 =\frac23+\frac34 =\frac{17}{12}.\]

The continuous term and the point contribution are evaluated exactly according to their mathematical type.

Null sets depend on the measure

An assertion that holds “almost everywhere” is incomplete until the measure is named. In the previous example,

\[\lambda(\{1/2\})=0,\qquad \mu_\alpha(\{1/2\})=3.\]

Changing a function at $x=1/2$ does not change its Lebesgue integral, but it changes its $\mu_\alpha$-integral. This is why geometric tolerances, discrete events, and concentrated constraints must not be discarded merely because they occupy zero Euclidean length.

Computational interpretation

The Lebesgue–Stieltjes model is appropriate when an accumulated quantity has a meaningful cumulative function $\alpha$. It preserves the semantic difference between density and jump, while allowing both to participate in the same convergence theorems and $L^p$ spaces.

For CAD and simulation software, this suggests a clean mathematical architecture: retain continuous field contributions as densities, retain discrete feature or constraint contributions as atoms, and combine them through the measure rather than through sampling-dependent approximations.