This chapter connects the measure-theoretic definition to familiar calculus. The central issue is not symbolic manipulation, but identifying the continuous, atomic, and singular parts of the integrator before applying a formula.

Absolutely continuous integrators

Let $\alpha$ be absolutely continuous on $[a,b]$. Then $\alpha’$ exists almost everywhere, $\alpha’\in L^1$, and

\[\alpha(x)=\alpha(a)+\int_a^x\alpha'(t)\,dt.\]

The associated Stieltjes measure satisfies

\[d\mu_\alpha(x)=\alpha'(x)\,dx.\]

Therefore, for every nonnegative or $\mu_\alpha$-integrable $f$,

\[\int_{(a,b]} f(x)\,d\alpha(x) =\int_a^b f(x)\alpha'(x)\,dx.\]

This reduction is exact only for the absolutely continuous part. It cannot recover atomic or singular continuous mass.

Jumps are point masses

For a right-continuous nondecreasing $\alpha$, define its jump at $c$ by

\[\Delta\alpha(c)=\alpha(c)-\alpha(c-).\]

Then

\[\mu_\alpha(\{c\})=\Delta\alpha(c).\]

If $\alpha$ has an absolutely continuous part and jumps $a_k\ge0$ at points $x_k$, then

\[\int f\,d\alpha =\int f(x)\alpha'_{\mathrm{ac}}(x)\,dx +\sum_k a_k f(x_k) +\int f\,d\alpha_{\mathrm{sc}},\]

where the last term is the singular continuous contribution. The decomposition is conceptual: a general monotone function may contain all three parts.

Fundamental theorem in measure form

If $f\in L^1([a,b])$, define

\[F(x)=\int_a^x f(t)\,dt.\]

Then $F$ is absolutely continuous and $F’(x)=f(x)$ almost everywhere. Conversely, every absolutely continuous $F$ has this representation with $f=F’$ almost everywhere.

The almost-everywhere qualification is sharp. For the Cantor function $C$, $C’=0$ almost everywhere, but $C(1)-C(0)=1$. Its change is carried by a singular measure and cannot be reconstructed from $C’\,dx$.

Integration by parts

When $f$ and $\alpha$ are absolutely continuous,

\[\int_a^b f(x)\,d\alpha(x) =f(b)\alpha(b)-f(a)\alpha(a) -\int_a^b\alpha(x)\,df(x),\]

which is the classical formula

\[\int_a^b f(x)\alpha'(x)\,dx =f(b)\alpha(b)-f(a)\alpha(a) -\int_a^b\alpha(x)f'(x)\,dx.\]

For functions of bounded variation with jumps, the endpoint and one-sided-value convention must be fixed. Under a common càdlàg convention, the product rule includes the contribution of simultaneous jumps:

\[\Delta(f\alpha)(x) =f(x-)\Delta\alpha(x) +\alpha(x-)\Delta f(x) +\Delta f(x)\Delta\alpha(x).\]

Thus copying the continuous integration-by-parts formula into a discontinuous setting can omit a jump term. The safe procedure is to state the interval convention and derive the formula from the measure decomposition.

Change of variables as pushforward

Let $T:X\to Y$ be measurable and let $T_#\mu$ denote the pushforward measure,

\[(T_\#\mu)(B)=\mu(T^{-1}(B)).\]

Then for every nonnegative measurable $g$,

\[\int_Y g(y)\,d(T_\#\mu)(y) =\int_X g(T(x))\,d\mu(x).\]

This identity is the measure-theoretic core of change of variables. Jacobian factors arise when the pushforward has a density with respect to a reference measure. In one dimension, a continuously differentiable strictly increasing map $T$ gives the familiar formula

\[\int_{T(a)}^{T(b)}g(y)\,dy =\int_a^b g(T(x))T'(x)\,dx.\]

The pushforward statement remains valid even when no smooth inverse or density exists.

Worked example: distributed and concentrated response

On $[0,1]$, let

\[\alpha(x)=x^2+4\mathbf 1_{[3/4,\,1]}(x), \qquad f(x)=1+x.\]

Then $d\alpha=2x\,dx+4\delta_{3/4}$, and

\[\begin{aligned} \int_{(0,1]}(1+x)\,d\alpha(x) &=\int_0^1(1+x)2x\,dx +4\left(1+\frac34\right)\\ &=\left[x^2+\frac{2}{3}x^3\right]_0^1+7\\ &=\frac{26}{3}. \end{aligned}\]

Approximating the jump by a narrow smooth peak can reproduce this value only if the numerical scheme resolves the peak and preserves its area. The Stieltjes representation expresses the point contribution exactly.

Differentiating parameterized integrals

Let

\[I(t)=\int_XF(x,t)\,d\mu(x).\]

If $F(\cdot,t)$ is measurable, $F(x,\cdot)$ is differentiable near $t_0$, and an integrable $g$ satisfies

\[\left|\partial_tF(x,t)\right|\le g(x)\]

near $t_0$, then

\[I'(t_0)=\int_X\partial_tF(x,t_0)\,d\mu(x).\]

This theorem gives a rigorous basis for sensitivities of mass properties, objective functions, and residual norms. If the measure itself depends on $t$, an additional derivative of the measure is required; differentiating only the integrand is incomplete.

Computational interpretation

The practical rule is to decompose before evaluating:

  • use a density integral for the absolutely continuous part;
  • use exact weighted evaluations for atoms;
  • retain a genuine measure representation for singular continuous mass;
  • use pushforwards for parameter transformations;
  • verify domination before interchanging differentiation and integration.

This preserves concentrated geometric or physical effects and makes the numerical implementation correspond to the actual mathematical object.