Proposition 11.5.1. Let $[a, b] \subset \real$, $E, F, H$ be TVSs over $K \in \RC$, and $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous linear map.
Let $G: [a, b] \to F$ and $[c, d] \subset [a, b]$ such that $G$ is continuous at $c$ and $d$, then for any $x \in E$, $x \cdot \one_{[c, d]}\in RS([a, b], G)$, and
\[\int_{a}^{b} x \cdot \one_{[c, d]}dG = x \cdot [G(d) - G(c)]\]
Proof. Assume without loss of generality that $a < c \le d < b$. Let $U \in \cn_{H}(0)$, then there exists $V \in \cn_{F}(0)$ such that $xV \subset U$. By continuity of $G$, there exists $\delta > 0$ such that $G((c - \delta, c]) - G(c) \subset V$ and $G([d, d + \delta)) - G(d) \subset V$. In which case, for any tagged partition $(P = \bracsn{x_j}_{0}^{n}, t = \seqf{t_j})$ that contains $\bracs{c - \delta, c, d, d + \delta}$,
\begin{align*}S(Q, t, x \cdot \one_{[c, d]}, G)&= x\sum_{a < x_j \le c - \delta}\one_{[c, d]}(t_{j})[G(x_{j}) - G(x_{j - 1})] \\&+ x \sum_{c - \delta < x_j \le c}\one_{[c, d]}(t_{j})[G(x_{j}) - G(x_{j - 1})] \\&+ x \sum_{c < x_j \le d}\one_{[c, d]}(t_{j})[G(x_{j}) - G(x_{j-1})] \\&+ x \sum_{d < x_j \le d + \delta}\one_{[c, d]}(t_{j})[G(x_{j}) - G(x_{j - 1})] \\&+ x \sum_{d + \delta < x_j \le b}\one_{[c, d]}(t_{j})[G(x_{j}) - G(x_{j - 1})] \\&= x \sum_{c - \delta \le x_j \le c}\one_{[c, d]}(t_{j})[G(x_{j}) - G(x_{j - 1})] \\&+ x \cdot [G(d) - G(c)] \\&+ x \sum_{d < x_j \le d + \delta}\one_{[c, d]}[G(x_{j}) - G(x_{j - 1})] \\&\in G(d) - G(c) + xG([c - \delta, c]) + xG([d, d + \delta]) \\&\subset x \cdot [G(d) - G(c)] + xV + xV \subset [G(d) - G(c)] + U + U\end{align*}
$\square$