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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:

  1. Rudin didn't write Theorem A in his book. Why?
  2. 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?
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    $\begingroup$ Rudin ‘s theorem is nominally more general. $\endgroup$ – Charlie Frohman Feb 25 at 13:08
  • $\begingroup$ @CharlieFrohman But I think that Theorem A is more general than Rudin's corollary about $\phi$. $\endgroup$ – tchappy ha Feb 25 at 13:22
  • $\begingroup$ And I think Rudin's corollary is more general than Theorem A about $f$. $\endgroup$ – tchappy ha Feb 26 at 2:13

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