Thank you very much, Saaqib Mahmood, for your text.
I copied and pasted it:
Theorem 6.19 on pp.132-133:
Suppose $\varphi$ is a strictly increasing continuous function that maps an interval $[ A, B]$ onto $[ a, b]$. Suppose $\alpha$ is monotonically increasing on $[ a, b]$ and $f \in \mathscr{R}(\alpha)$ on $[a, b]$. Define $\beta$ and $g$ on $[ A, B]$ by $$ \beta(y) = \alpha \left( \varphi(y) \right), \qquad g(y) = f \left( \varphi(y) \right). \tag{36} $$ Then $g \in \mathscr{R}(\beta)$ and $$ \int_A^B g \ \mathrm{d} \beta = \int_a^b f \ \mathrm{d} \alpha. \tag{37} $$
Corollary:
Let us note the following special case:
Take $\alpha(x) = x$. Then $\beta=\phi$. Assume $\phi' \in \mathscr{R}$ on $[A, B]$. If Theorem 6.17 is applied to the left side of (37), we obtain
$$\int_a^b f(x) \ \mathrm{d} x = \int_A^B f(\phi(y)) \phi'(y) \mathrm{d} y. \tag{39}$$
In many other books, there is the following theorem instead of the above corollay.
Theorem A:
Suppose that $\phi$ is a differentiable function on $[A, B]$.
Suppose that $\phi([A, B]) \subset [a, b]$.
Suppose that $\phi' \in \mathscr{R}$ on $[A, B]$.
Suppose $f$ is continuous on $[a, b]$.
Then $f(\phi(y)) \phi'(y) \in \mathscr{R}$ on $[A, B]$ and
$$ \int_A^B f(\phi(y)) \phi'(y) \ \mathrm{d} y = \int_{\phi(A)}^{\phi(B)} f(x) \ \mathrm{d} x.$$
My questions are here:
- Rudin didn't write Theorem A in his book. Why?
- About $\phi$, I think Theorem A is more general than Rudin's corollary. About $f$, I think Rudin's corollary is more general than Theorem A. Which theorem is more useful for applications?
