Darboux criterion for Riemann–Stieltjes integrability with a nondecreasing integrator

For every partition $P$, $U_\alpha(f,P)-L_\alpha(f,P)=\sum_{i<n}(M_i-m_i)\Delta_i\alpha=\sum_{i<n}\omega_i(f)\Delta_i\alpha$. Refinement can only decrease the upper sum and increase the lower sum, because each refined supremum is no larger and each refined infimum no smaller than its coarse counterpart.

Suppose first that the mesh-limit integral is $I$. Given $\varepsilon>0$, choose a partition $P$ fine enough that every tagged sum over $P$ is within $\varepsilon/4$ of $I$. In each subinterval choose tags whose values approach its supremum and infimum within a common error small enough, using [L5] and the finite number of intervals. The two resulting tagged sums differ by more than $U_\alpha(f,P)-L_\alpha(f,P)-\varepsilon/2$, but by less than $\varepsilon/2$ through $I$; hence $U_\alpha(f,P)-L_\alpha(f,P)<\varepsilon$.

Mesh integrability also forces continuity of $f$ at every discontinuity $c$ of $\alpha$. By [L6], the total increment of $\alpha$ across every sufficiently small interval straddling $c$ is bounded below by a fixed positive number. Complete such an interval to an arbitrarily fine partition and keep every other tag fixed. Tagging the straddling interval first at $c$ and then at an arbitrary point $x$ in that interval changes the sum by $(f(x)-f(c))\Delta\alpha$. Both sums must approach the same mesh limit, so $f(x)\to f(c)$ as $x\to c$. The same one-sided argument applies at an endpoint.

Conversely assume both stated conditions. The lower sums have a supremum $I_-$ and the upper sums an infimum $I_+$, with $I_-\le I_+$. Step 1.1 and condition 2 force $I_-=I_+=:I$. Given $\varepsilon>0$, choose $P_0$ with Darboux gap below $\varepsilon/3$. Around each of its finitely many interior points $c$, choose a small neighborhood as follows: if $\alpha$ is continuous at $c$, make the variation of $\alpha$ there so small that twice the bound on $|f|$ times that variation is below the allotted error; if $\alpha$ is discontinuous at $c$, condition 1 and [L7] make the oscillation of $f$ there so small that its product with $\alpha(b)-\alpha(a)$ is below the allotted error. Choose the neighborhoods disjoint and divide the error among their finite number.

Let $Q$ now have mesh smaller than all those neighborhood radii and let $R=P_0\vee Q$. A tagged sum on $R$ lies between $L_\alpha(f,P_0)$ and $U_\alpha(f,P_0)$. Comparing a sum on $Q$ with one on $R$, the intervals of $Q$ that do not cross a point of $P_0$ contribute at most the Darboux gap. Each crossing interval lies in one chosen neighborhood: its refinement error is bounded either by $2\sup|f|$ times the local variation of $\alpha$, or by the local oscillation of $f$ times the total variation $\alpha(b)-\alpha(a)$. The choices in step 2.1 make the sum of all crossing errors below $2\varepsilon/3$. Hence every sufficiently fine tagged sum lies within $\varepsilon$ of $I$.

Steps 1.2–1.3 prove necessity, steps 2.1–3.1 prove sufficiency, and step 1.1 proves the weighted-oscillation formulation. When $\alpha$ is continuous, condition 1 is vacuous.