Define \(f,\alpha\in\mathcal B([-2,2])\) by \[ f(x):=\begin{cases} -1,&x\in[-2,0),\\ 3,&x\in[0,2], \end{cases} \qquad \alpha(x):=\begin{cases} -2,&x\in[-2,0],\\ 1,&x\in(0,2]. \end{cases} \] Determine whether \(f\) is Riemann–Stieltjes integrable with respect to \(\alpha\) over \([-1,1]\). If it is, evaluate \[\int_{-1}^{1}f(x)\,d\alpha(x).\]
Solution
Goal: Determine whether \(f\) is Riemann–Stieltjes integrable w.r.t. \(\alpha\) on \([-1,1]\), where \(f = -1\) on \([-2,0)\), \(3\) on \([0,2]\); \(\alpha = -2\) on \([-2,0]\), \(1\) on \((0,2]\). If so, evaluate \(\int_{-1}^1 f\,d\alpha\).
1. The restriction to \([-1,1]\): \(f(x) = -1\) on \([-1,0)\), \(f(x) = 3\) on \([0,1]\); \(\alpha(x) = -2\) on \([-1,0]\), \(\alpha(x) = 1\) on \((0,1]\).Proof: restrict the given definitions to \([-1,1]\).
2. \(f \in \mathcal R(\alpha)\) on \([-1,1]\) and the integral equals \(9\).
2.1. Use a partition that contains the jump point.Proof: take the partition \(P=\{-1,0,1\}\). If \(Q\) is any refinement of \(P\), then \(\alpha\) is constant on every subinterval of \(Q\) except the first subinterval \([0,q]\) immediately to the right of \(0\). On that interval\[ \alpha(q)-\alpha(0)=1-(-2)=3. \]
2.2. Every tagged Riemann–Stieltjes sum for every refinement of \(P\) equals \(9\).Proof: on the exceptional interval \([0,q]\) one has \(f\equiv3\), including at \(0\). Hence its contribution is\[ 3\,[\alpha(q)-\alpha(0)]=3\cdot3=9, \]while every other interval contributes \(0\) because its \(\alpha\)-increment is \(0\). Thus the sum is independent of both the refinement and the tags.
2.3. Apply the Riemann–Stieltjes refinement criterion.Proof: given any \(\varepsilon>0\), the partition \(P\) above has the property that every tagged sum over every refinement of \(P\) equals \(9\), hence differs from \(9\) by less than \(\varepsilon\). Therefore\[ \boxed{f\in\mathcal R(\alpha),\qquad \int_{-1}^1 f\,d\alpha=9.} \]