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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 $$ \int_A^B g \ \mathrm{d} \beta = \int_a^b f \ \mathrm{d} \alpha. \tag{37} $$

Rudin’s proof:

To each partition $P = \{ \ x_0, \ldots, x_n \ \}$ of $[a, b]$ corresponds a partition $Q = \{ \ y_0, \ldots, y_n \ \}$ of $[ A, B]$, so that $x_i = \varphi \left( y_i \right)$. All partitions of $[A, B]$ are obtained in this way. Since the values taken by $f$ on $\left[ x_{i-1}, x_i \right]$ are exactly the same as those taken by $g$ on $\left[ y_{i-1}, y_i \right]$, we see that $$ \tag{38} U(Q, g, \beta) = U(P, f, \alpha), \qquad L(Q, g, \beta) = L(P, f, \alpha). $$ Since $f \in \mathscr{R}(\alpha)$, $P$ can be chosen so that both $U(P, f, \alpha)$ and $L(P, f, \alpha)$ are close to $\int f \ \mathrm{d} \alpha$. Hence (38), combined with Theorem 6.6, shows that $g \in \mathscr{R}(\beta)$ and that (37) holds. This completes the proof.

Question I don’t understand the last step, especially (37) holds: Hence (38), combined with Theorem 6.6, shows that $g \in \mathscr{R}(\beta)$ and that (37) holds.

user264745
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    It might be good to include the statement of Theorem $6.6$. – podiki Dec 19 '21 at 19:20
  • Theorem 6.6: $f \in \mathscr{R}$ on $[a,b]$ if and only if $\forall \epsilon \gt 0, \exists$ a partition P such that $U(P,f,\alpha) - L(P,f,\alpha) \lt \epsilon$ – user264745 Dec 19 '21 at 19:27

2 Answers2

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As $f\in\mathscr{R}(\alpha)$, by Theorem 6.6 there exists a partition $P$ consisting of $x_i\in[a,b]$ such that

$$U(P,f,\alpha)-L(P,f,\alpha) < \epsilon\ . $$

But $Q$ defined by the values $\varphi(y_i)=x_i$ is a partition of $[A,B]$ and by the equality (38) we have

$$U(Q,g,\beta)-L(Q,g,\beta) < \epsilon\ .$$

Thus by the other direction of Theorem 6.6 we have that $g\in\mathscr{R}(\beta)$.

To see that they are equal, you just need to see that

$$\sup_{P}L(P,f,\alpha) = \sup_Q L(Q,g,\beta)\ .$$

Let $P=\{x_i\}$ be a partition of $[a,b]$ and define $Q=\{\varphi^{-1}(x_i)\}$. This is a partition of $[A,B]$ so we have

$$ L(P,f,\alpha) = L(Q,g,\beta)\leq \sup_{Q}L(Q,g,\beta)\ .$$

Now we take the $\sup$ over all such $P$ to see

$$\sup_{P}L(P,f,\alpha) \leq \sup_Q L(Q,g,\beta)\ .$$

Conversely, suppose $Q=\{y_i\}$ is a partition of $[A,B]$ and we can define $P=\{\varphi(y_i)\}$ which is a partition of $[a,b]$. Then we have

$$L(Q,g,\beta)=L(P,f,\alpha)\leq \sup_{P} L(P,f,\alpha)\ .$$ Taking $\sup$ over all such $Q$ gives

$$\sup_{Q}L(Q,g,\beta) \leq \sup_P L(P,f,\alpha)\ .$$

podiki
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  • Thank you for the answer. If you see my this post(comment section of both post and 1 answer), https://math.stackexchange.com/q/4337475/861687. We’ll find out, I’m exactly stuck at that point where everyone thinks it’s obvious. – user264745 Dec 19 '21 at 20:40
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    @user264745 I edited my answer. It is more than just that there is a bijection between partitions of $[a,b]$ and $[A,B]$, it is that $\varphi$ (being invertible) gives an explicit correspondence between them that preserves the values of $L$ and $U$. – podiki Dec 19 '21 at 20:52
  • Mannn..... how did I miss that((37) holds) proof? It is the most logical direction to go. This is not my first time missing “similar pattern(involving the concept of sup, inf, and partition)” of proofs. This is the post I’m talking about https://math.stackexchange.com/q/4320461/861687. Anyway here’s the summary of this answer. – user264745 Dec 20 '21 at 10:39
  • $\forall P \in \mathcal{P}, \exists Q \in \mathcal{Q}$ such that $L(P,f,\alpha)=L(Q,g,\beta)$, where $\mathcal{P}$ & $\mathcal{Q}$ denotes the set of all partition of $[a,b]$ and $[A,B]$, respectively. $\forall \epsilon \gt 0, \exists Q$, a partition of $[A,B]$ (existence of partition $Q$ depends on the hypothesis of the problem, $f \in \mathscr{R} (\alpha)$, to be more precise) such that $U(Q,g,\beta)-L(Q,g,\beta) \lt \epsilon$. Hence $g \in \mathscr{R} (\beta)$. – user264745 Dec 20 '21 at 10:39
  • To prove: $\sup_{Q}L(Q,g,\beta) = \sup_P L(P,f,\alpha)$. Let $P$ be a partition of $[a,b]$, $\exists Q$ a partition of $[A,B]$ such that $L(P,f,\alpha) = L(Q,g,\beta)\leq \sup_{Q}L(Q,g,\beta)$. Now the inequality, $L(P,f,\alpha) \leq \sup_{Q}L(Q,g,\beta)$, holds for all $P \in \mathcal{P}$. Thus $\sup_{P}L(P,f,\alpha) \leq \sup_Q L(Q,g,\beta)$. Converse of this inequality, is similar. – user264745 Dec 20 '21 at 10:39
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Theorem 6.6 says the following:

$f \in \mathscr{R}(\alpha)$ if and only if for every $\epsilon > 0$ there exists a partition $P$ such that

$$U(P,f,\alpha) - L(P,f,\alpha) < \epsilon.$$

Put another way: $f$ is integrable iff you can make the upper and lower Reimann sums arbitrarily close by choosing the right partition. This is what he's referring to when he says that $U$ and $L$ can be made close to $\int f d\alpha$. The fact that partitions on $[A, B]$ can be identified with partitions on $[a, b]$, and that the associated Reimann sums are equal, is what ties the fates of $f$ and $g$ together. All of the information about integrability is captured in the partitions and associated Reimann sums.

BlackboardN
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