For \(\alpha\) increasing on \([a,b]\), a bounded \(f\) is Riemann–Stieltjes integrable with respect to \(\alpha\) if and only if for every \(\varepsilon>0\) there is a partition \(P\) of \([a,b]\) with \(U(P,f,\alpha)-L(P,f,\alpha)<\varepsilon\). If \(f\) is integrable with respect to \(\alpha\), and \(\alpha\) is differentiable with \(\alpha'\) Riemann integrable, then \(\int_a^bf\,d\alpha=\int_a^bf(x)\alpha'(x)\,dx\). Integration by parts: if \(f\) is integrable with respect to \(\alpha\), then \(\alpha\) is integrable with respect to \(f\) and \(\int_a^bf\,d\alpha=f(b)\alpha(b)-f(a)\alpha(a)-\int_a^b\alpha\,df\). The fundamental theorem of calculus: if \(f\) is Riemann integrable on \([a,b]\) and \(F\) is differentiable on \([a,b]\) with \(F'=f\), then \(\int_a^bf=F(b)-F(a)\); and if \(f\) is continuous, then \(x\mapsto\int_a^xf\) is differentiable with derivative \(f\). For continuous \(f\), \(\int_a^bf=f(c)(b-a)\) for some \(c\in[a,b]\). Lebesgue’s criterion: a bounded function on \([a,b]\) is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero.
Riemann's condition for Riemann--Stieltjes integrability (Tag T-RA-WORKSHOP-D6-5-1)
Let \(a<b\), let \(\alpha\colon[a,b]\to\RR\) be monotonically increasing, and let \(f\colon[a,b]\to\RR\) be bounded. For a partition \(P=\theset{a=x_0<x_1<\cdots<x_k=b}\) of \([a,b]\), put \(\Delta\alpha_i\coloneqq\alpha(x_i)-\alpha(x_{i-1})\) and \[ U(P,f,\alpha)\coloneqq\sum_{i=1}^k\qty{\sup_{[x_{i-1},x_i]}f}\Delta\alpha_i,\qquad L(P,f,\alpha)\coloneqq\sum_{i=1}^k\qty{\inf_{[x_{i-1},x_i]}f}\Delta\alpha_i . \] Then \(f\in\mathcal R(\alpha)\), that is, \(f\) is Riemann–Stieltjes integrable with respect to \(\alpha\) on \([a,b]\), if and only if for every \(\varepsilon>0\) there exists a partition \(P\) of \([a,b]\) such that \[ U(P,f,\alpha)-L(P,f,\alpha)<\varepsilon. \] [@Rud76].
Riemann--Stieltjes integration by parts (Tag T-RA-WORKSHOP-D6-5-2)
Let \(a<b\) and let \(f,\alpha\colon[a,b]\to\RR\). Write \(f\in\mathcal R(\alpha)\) if there is \(A\in\RR\) such that for every \(\varepsilon>0\) there is a partition \(P_\varepsilon\) of \([a,b]\) with \[ \abs{\sum_{i=1}^k f(t_i)\qty{\alpha(x_i)-\alpha(x_{i-1})}-A}<\varepsilon \] for every partition \(P=\theset{a=x_0<\cdots<x_k=b}\) refining \(P_\varepsilon\) and every choice of \(t_i\in[x_{i-1},x_i]\); then \(\int_a^b f\,d\alpha\coloneqq A\). If \(f\in\mathcal R(\alpha)\) on \([a,b]\), then \(\alpha\in\mathcal R(f)\) on \([a,b]\) and \[ \int_a^b f\,d\alpha=f(b)\alpha(b)-f(a)\alpha(a)-\int_a^b\alpha\,df. \] [@Apo74].
Reduction of a Stieltjes integral to a Riemann integral (Tag T-RA-WORKSHOP-D6-5-3)
Let \(a<b\), let \(\alpha\colon[a,b]\to\RR\) be monotonically increasing and differentiable with \(\alpha'\) Riemann integrable on \([a,b]\), and let \(f\colon[a,b]\to\RR\) be bounded. Then \(f\in\mathcal R(\alpha)\) on \([a,b]\) if and only if \(f\alpha'\) is Riemann integrable on \([a,b]\), and in that case \[ \int_a^b f\,d\alpha=\int_a^b f(x)\alpha'(x)\,dx. \] [@Rud76].
First fundamental theorem of calculus (Tag T-RA-WORKSHOP-D6-5-4)
Let \(a<b\), let \(f\colon[a,b]\to\RR\) be Riemann integrable on \([a,b]\), and define \(F\colon[a,b]\to\RR\) by \[ F(x)\coloneqq\int_a^x f(t)\,dt. \] Then \(F\) is continuous on \([a,b]\). If \(f\) is continuous at \(x_0\in[a,b]\), then \(F\) is differentiable at \(x_0\), one-sidedly if \(x_0\in\theset{a,b}\), and \(F'(x_0)=f(x_0)\). [@Rud76].
Mean value theorem for integrals (Tag T-RA-WORKSHOP-D6-5-5)
Let \(a<b\) and let \(f\colon[a,b]\to\RR\) be continuous. Then there exists \(c\in(a,b)\) such that \[ \frac{1}{b-a}\int_a^b f(x)\,dx=f(c). \] [@Apo74].
Lebesgue criterion for Riemann integrability (Tag PR-TDH2A)
Let \(a<b\) and let \(f\colon[a,b]\to\RR\) be bounded, with set of discontinuities \(D_f\subseteq[a,b]\). Then \(f\) is Riemann integrable on \([a,b]\) if and only if \(D_f\) has Lebesgue measure \(0\).