Prove : \(f \in \mathcal{R}(\alpha)\) on \([a,b]\) if and only if for any \(a <c<b\), \(f \in \mathcal{R}(\alpha)\) on \([a,c]\) and on \([c,b]\). In addition, if either condition holds, then we have that \[\int_a^c fd\alpha + \int_c^b fd\alpha = \int_a^b fd\alpha.\]
Solution
1. (\(\Rightarrow\)) If \(f \in \mathcal R(\alpha)\) on \([a,b]\) then \(f \in \mathcal R(\alpha)\) on \([a,c]\) and on \([c,b]\).
given partitions \(P_1, P_2\) of \([a,c], [c,b]\), their union \(P\) is a partition of \([a,b]\) with \(U(f,P,\alpha) - L(f,P,\alpha) = \big(U(f,P_1,\alpha) - L(f,P_1,\alpha)\big) + \big(U(f,P_2,\alpha) - L(f,P_2,\alpha)\big)\) (the \(\alpha\)-increments on the two subintervals add up to those on \([a,b]\); note \(f\) is bounded on \([a,b]\) by integrability).
Since integrability on \([a,b]\) lets \(U - L \to 0\) along such partitions, each subinterval’s difference tends to \(0\), giving integrability on each.
2. (\(\Leftarrow\)) If \(f \in \mathcal R(\alpha)\) on \([a,c]\) and on \([c,b]\), then \(f \in \mathcal R(\alpha)\) on \([a,b]\).
2.1. Given \(\eps > 0\), choose partitions \(P_1\) of \([a,c]\) and \(P_2\) of \([c,b]\) with \(U(f,P_1,\alpha) - L(f,P_1,\alpha) < \eps/2\) and \(U(f,P_2,\alpha) - L(f,P_2,\alpha) < \eps/2\).
integrability on each subinterval.
2.2. \(P = P_1 \cup P_2\) is a partition of \([a,b]\) with \(U(f,P,\alpha) - L(f,P,\alpha) < \eps\).
the upper/lower sums split over the two subintervals as in 1 (the boundary point \(c\) contributes its \(\Delta\alpha\) to exactly one of the two sums), so \(U - L = (U_1 - L_1) + (U_2 - L_2) < \eps\).
2.3. Q.E.D.
2 shows \(U - L\) can be made arbitrarily small, i.e. integrability on \([a,b]\) (Cauchy criterion).
3. Additivity: \(\int_a^c f\,d\alpha + \int_c^b f\,d\alpha = \int_a^b f\,d\alpha\).
for the partitions of 2, Riemann–Stieltjes sums satisfy \(S(P) = S(P_1) + S(P_2)\); taking \(P_1, P_2\) with mesh making all three sums converge to the respective integrals gives \(\int_a^b = \int_a^c + \int_c^b\) (both sides are limits of the same telescoping sums).
4. Q.E.D.
1, 2, 3 establish both directions and the additivity formula.