# About Theorem 6.19 on pp.132-133 in “Principles of Mathematical Analysis” by Walter Rudin

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?
• Rudin ‘s theorem is nominally more general. – Charlie Frohman Feb 25 at 13:08
• @CharlieFrohman But I think that Theorem A is more general than Rudin's corollary about $\phi$. – tchappy ha Feb 25 at 13:22
• And I think Rudin's corollary is more general than Theorem A about $f$. – tchappy ha Feb 26 at 2:13