Proposition 13.5.3: Change of Variables

Proposition 13.5.3 (Change of Variables).label Let $[a, b], [c, d] \subset \real$, $E, F, H$ be locally convex spaces over $K \in \RC$, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, $\gamma \in C([a, b]; F)$ be a path, and $\varphi: C([c, d]; [a, b])$ be non-decreasing with $\varphi(c) = a$ and $\varphi(d) = b$, then for any $f \in PI([a, b], \gamma; E)$, $f \in PI([c, d], \gamma \circ \varphi; E)$, and

\[\int_{\gamma} f = \int_{\gamma \circ \varphi}f\]

Proof. Since $\varphi(c) = a$, $\varphi(d) = b$, and $\varphi$ is continuous, it is surjective. As $\varphi$ is also non-decreasing, for any tagged partition $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_{t}([a, b])$, there exists a tagged partition $(Q = \seqfz{y_j}, d = \seqf{d_j}) \in \scp_{t}([c, d])$ such that $\varphi(y_{j}) = x_{j}$ for each $0 \le j \le n$ and $\varphi(d_{j}) = c_{j}$ for each $1 \le j \le n$. In addition,

\begin{align*}S(P, c, f \circ \gamma, \gamma)&= \sum_{j = 1}^{n} f \circ \gamma(c_{j})[\gamma(x_{j}) - \gamma(x_{j - 1})] \\&= \sum_{j = 1}^{n} f \circ \gamma \circ \varphi (d_{j})[\gamma \circ \varphi(y_{j}) - \gamma \circ \varphi(y_{j-1})] \\&= S(Q, d, f \circ \gamma \circ \varphi, \gamma \circ \varphi)\end{align*}

Therefore if $f \in PI([a, b], \gamma; E)$, then $f \in PI([c, d], \gamma \circ \varphi; E)$, with $\int_{\gamma} f = \int_{\gamma \circ \varphi}f$.$\square$

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