Can Rudin's PMA Theorem 6.17 be proven by using Riemann (as opposed to Darboux) integrals?

Question

Throughout this post I refer to these as "Darboux integrals/integration" and to these as "Riemann integrals/intnegration". It is a standard result of Analysis that these notions of integrability coincide.

---

I've tried (and failed) looking for a book that introduces integration in terms of Riemann integrals: Rudin's POMA, Tao's Analysis I, Spivak's Calculus and Abbott's Understanding Analysis all use Darboux integration. Wikipedia explains the reason in its article on Darboux integration:

"The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral."

Itching my curiosity I've shown that some elementary theorems on integration can be proven directly using Riemann integrability (such as that $\int(f+g)=\int f + \int g$, as well as $c\int f = \int cf$). Currently I'm stuck trying to prove Rudin's 6.17 and hence I'm asking for a hand.

---

Theorem 6.17: Assume $\alpha$ increases monotonically on $[a,b]$ and $\alpha' \in \mathcal{R}$ (i.e. $\alpha'$ is Darboux integrable). Let $f$ be a bounded real function on $[a,b]$. Then $f\in \mathcal{R}(\alpha)$ if and only if $f\alpha'\in\mathcal{R}$. In that case $$\int_a^bf\ d\alpha = \int_a^bf(x)\alpha'(x)\ dx.$$

---

Can 6.17 be shown by using Riemann integration?

To be clearer, showing the result by directly employing the criterion in the definition of Riemann integrability, as opposed to using other (equivalent) criteria.

There would be no point in using the criterion on the definition of Darboux integrability, nor to use, as Rudin did, the fact that a function is Riemann integrable if and only if for any $\epsilon>0$, there is some partition $P$ such that $$\left|U(f,\alpha,P)-L(f,\alpha,P)\right|<\epsilon.$$

Answer 1

Here is a proof (without refering to Darboux integrals) of the reverse implication of 6.17 that if $f \alpha'$ is Riemann integrable on $[a,b]$ then $f$ is Riemann-Stieltjes integrable with respect to the integrator $\alpha$ on $[a,b]$ and

$$\int_a^b f \, d\alpha = \int_a^b f(x) \alpha'(x)\, dx$$

The objective is to show that for any $\epsilon >0$, there exists a partition $P_\epsilon$ such that for any refinement

$$P = \{x_0,x_1,\ldots,x_n:a = x_0 < x_1 < \ldots <x_n = b\}$$

and any choice of intermediate points $\xi_j \in [x_{j-1},x_j]$, we have

$$\left|S(P,f,\alpha,\{\xi_j\})- \int_a^bf(x) \alpha'(x) \, dx\right|= \left|\sum_{j=1}^nf(\xi_j)[\alpha(x_j) - \alpha(x_{j-1})]- \int_a^bf(x) \alpha'(x) \, dx\right|< \epsilon$$

By the mean value theorem, there exists $\eta_j \in (x_{j-1},x_j)$ for $j=1,2,\ldots,n$ such that

$$\sum_{j=1}^nf(\xi_j)[\alpha(x_j) - \alpha(x_{j-1})]= \sum_{j=1}^nf(\xi_j)\alpha'(\eta_j) (x_j - x_{j-1})$$

Thus,

$$\left|\sum_{j=1}^nf(\xi_j)[\alpha(x_j) - \alpha(x_{j-1})]- \int_a^bf(x) \alpha'(x) \, dx\right|\\= \left|\sum_{j=1}^nf(\xi_j)\alpha'(\xi_j)(x_j-x_{j-1})- \int_a^bf(x) \alpha'(x) \, dx + \sum_{j=1}^nf(\xi_j)\alpha'(\eta_j)(x_j-x_{j-1})- \sum_{j=1}^nf(\xi_j)\alpha'(\xi_j)(x_j-x_{j-1})\right|\\ \leqslant \underbrace{\left|\sum_{j=1}^nf(\xi_j)\alpha'(\xi_j)(x_j-x_{j-1})- \int_a^bf(x) \alpha'(x) \, dx\right|}_{A}+ \underbrace{\sum_{j=1}^n |f(\xi_j)||\alpha'(\eta_j) - \alpha'(\xi_j)|(x_j - x_{j-1})}_{B}$$ Because $f\alpha'$ is Riemann integrable it follows that there exists a partition $P_A$ such that if $P$ is a refinement, then $$\tag{1}A= \left|\sum_{j=1}^nf(\xi_j)\alpha'(\xi_j)(x_j-x_{j-1})- \int_a^bf(x) \alpha'(x) \, dx\right|< \frac{\epsilon}{2}$$

Since $f$ is bounded, there exists $M>0$ such that $|f(x)| < M$ for all $x \in [a,b]$ and, hence

$$B = \sum_{j=1}^n |f(\xi_j)|[\alpha'(\eta_j) - \alpha'(\xi_j)|(x_j - x_{j-1}) < M \sum_{j=1}^n |\alpha'(\eta_j) - \alpha'(\xi_j)|(x_j - x_{j-1}) $$

It will be shown below that because $\alpha'$ is Riemann integrable, there exists a partition $P_B$ such that if $P$ is a refinement, then

$$\tag{2}\sum_{j=1}^n |\alpha'(\eta_j) - \alpha'(\xi_j)|(x_j - x_{j-1})<\frac{\epsilon}{2M},$$

whence,

$$\tag{3}B < M\sum_{j=1}^n |\alpha'(\eta_j) - \alpha'(\xi_j)|(x_j - x_{j-1}) < \frac{\epsilon}{2}$$

Hence, if P is a refinement of the common refinement $P_\epsilon = P_A \cup P_B$, then by (1) and (3) we have

$$\left|\sum_{j=1}^nf(\xi_j)[\alpha(x_j) - \alpha(x_{j-1})]- \int_a^bf(x) \alpha'(x) \, dx\right|=A+B < \epsilon,$$

and we have proved that $\int_a^b f \, d\alpha = \int_a^b f(x) \alpha'(x) \, dx$ as was to be shown.

The forward implication can be proved in a similar way.

Proof of (2)

Since $\alpha'$ is Riemann integrable, there exists a partition $P_B$ such that if $P$ is a refinement, then for any choice of intermediate points $t_j \in [x_{j-1},x_j]$, we have

$$\tag{4}\left|\sum_{j=1}^n \alpha'(t_j)(x_j - x_{j-1}) - \int_a^b \alpha'(x) \, dx\right|< \frac{\epsilon}{8M}$$

Denoting $M_j = \sup\{\alpha'(x): x \in [x_{j-1},x_j]\}$ and $m_j = \sup\{\alpha'(x): x \in [x_{j-1},x_j]\}$, by properties of the supremum and infimum there exist points $t_j', t_j''$ such that

$$M_j - \frac{\epsilon}{4M(b-a)} < \alpha'(t_j') \leqslant M_j, \quad m_j \leqslant \alpha'(t_j'') < m_j + \frac{\epsilon}{8(b-a)},$$

and, thus,

$$\sum_{j=1}^n (M_j - m_j)(x_j - x_{j-1}) < \sum_{j=1}^n \alpha'(t_j')(x_j- x_{j-1}) -\sum_{j=1}^n \alpha'(t_j'')(x_j- x_{j-1}) + \frac{\epsilon}{2M}\\ = \sum_{j=1}^n \alpha'(t_j')(x_j- x_{j-1})- \int_a^b\alpha'(x) \, dx + \int_a^b \alpha'(x) \, dx -\sum_{j=1}^n \alpha'(t_j'')(x_j- x_{j-1}) + \frac{\epsilon}{4M}$$

Hence, since the LHS is nonnegative,

$$\tag{5} \sum_{j=1}^n (M_j - m_j)(x_j - x_{j-1}) \\< \left|\sum_{j=1}^n \alpha'(t_j')(x_j- x_{j-1})- \int_a^b\alpha'(x) \, dx \right|+ \left| \int_a^b \alpha'(x) \, dx -\sum_{j=1}^n \alpha'(t_j'')(x_j- x_{j-1})\right|+ \frac{\epsilon}{4M}$$

Using (4), it follows from substitution into (5) that

$$ \sum_{j=1}^n (M_j - m_j)(x_j - x_{j-1})< \frac{\epsilon}{8M} + \frac{\epsilon}{8M} + \frac{\epsilon}{8M} = \frac{\epsilon}{2M}$$

Noting that $\xi_j, \eta_j \in [x_{j-1},x_j]$ and $|\alpha'(\xi_j)- \alpha'(\eta_j)| \leqslant M_j - m_j$, we obtain the result

$$\sum_{j=1}^n |\alpha'(\eta_j) - \alpha'(\xi_j)|(x_j - x_{j-1})\leqslant \sum_{j=1}^n (M_j - m_j)(x_j - x_{j-1})<\frac{\epsilon}{2M}$$

Answer 2

This is an alternative approach based on the following criterion of Riemann integrability:

Cauchy's condition for Riemann integrability: Let the function $f:[a, b] \to\mathbb{R} $ be bounded on $[a, b] $. Then $f$ is Riemann integrable on $[a, b] $ if and only if for every $\epsilon>0$ there exists a corresponding $\delta>0 $ (partition $P_{\epsilon} $ of $[a, b] $) such that $$|S(P_1,f)-S(P_2,f)|<\epsilon$$ where $S(P_1,f),S(P_2,f)$ are Riemann sums for $f$ over partitions $P_1,P_2$ and any choice of tags and $||P_1||<\delta,||P_2||<\delta$ ($P_1,P_2$ are finer than $P_{\epsilon} $ ie $P_{\epsilon} \subseteq P_1,P_{\epsilon}\subseteq P_2$).

In what follows we use the norm based definition/criterion (instead of refinement) due to less typing effort involved.

Let us state our hypotheses at the beginning. We are given a bounded function $f$ on $[a, b] $ as well as another function $\alpha$ whose derivative $\alpha'$ is Riemann integrable on $[a, b] $. We don't assume the monotone nature of $\alpha$ and further we don't assume Riemann integrability of $f$.

Let us assume $f\in\mathcal{R} (\alpha) $ with $I=\int_a^b f\, d\alpha$. Let $\epsilon>0$ and by the Riemann-Stieltjes integrability of $f$ with respect to $\alpha$ there exists a $\delta_1>0$ such that $$|S(P, f, \alpha) - I|<\frac{\epsilon} {2}\tag{1}$$ whenever $||P||<\delta_1$.

Let $M$ be a positive upper bound for $|f|$ over $[a, b] $ and applying Cauchy's condition on $\alpha'$ we get a $\delta_2>0$ such that $$|S(P_1,\alpha')-S(P_2,\alpha')|<\frac{\epsilon}{2M}\tag{2}$$ whenever $||P_1||<\delta_2,||P_2||<\delta_2$.

Let $\delta =\min(\delta_1,\delta_2)$ and $P$ be a partition of $[a, b] $ with $||P||<\delta$ and consider an arbitrary Riemann sum $$S(P, f\alpha') =\sum_{k=1}^n f(\xi_k) \alpha'(\xi_k) (x_k-x_{k-1})$$ for $f\alpha'$ over $P$. Next consider a specific Riemann sum $S(P, f, \alpha) $ for $f$ with respect to $\alpha$ over $P$ and tag points $\xi_k$ given by $$S(P, f, \alpha) =\sum_{k=1}^n f(\xi_k) (\alpha(x_k) - \alpha(x_{k-1}))=\sum_{k=1}^n f(\xi_k) \alpha'(\eta_k) (x_k-x_{k-1})$$ (via mean value theorem). We have \begin{align} |S(P, f\alpha') - I| &\leq |S(P, f\alpha') - S(P, f, \alpha) |+|S(P, f, \alpha) - I|\notag\\ &<\left|\sum_{k=1}^n f(\xi_k) (\alpha'(\xi_k) - \alpha'(\eta_k)) (x_k-x_{k-1})\right|+\frac{\epsilon} {2}\text{ (via (1))}\notag\\ &\leq M\sum_{k=1}^n |\alpha'(\xi_k) - \alpha'(\eta_k) |(x_k-x_{k-1})+\frac{\epsilon}{2}\notag\\ &<M\cdot\frac{\epsilon}{2M}+\frac{\epsilon}{2}\text{ (via (2))}\notag\\ &=\epsilon\notag \end{align} The last step which uses Cauchy's criterion needs a bit more explanation. Here we are taking both partitions $P_1,P_2$ equal to $P$ and two sets of tags $t_k, t'_k$ such that $t_k=\xi_k, t'_k=\eta_k$ if $\alpha'(\xi_k) \geq \alpha'(\eta_k) $ and $t_k=\eta_k, t'_k=\xi_k$ if $\alpha'(\xi_k) <\alpha'(\eta_k) $. This ensures that $$|\alpha'(\xi_k) - \alpha'(\eta_k) |=\alpha'(t_k) - \alpha'(t'_k) $$ for all $k$ and we get $\sum |\alpha'(\xi_k) - \alpha'(\eta_k)|(x_k-x_{k-1}) $ as a difference between two Riemann sums for $\alpha'$ over $P$. Note further that this uses the "only if" part of Cauchy's criterion which is an immediate consequence of definition of integral (since two Riemann sums are near the value of integral they are near to each other as well).

This proves that $f\alpha'$ is Riemann integrable with integral $I$.

To prove the reverse implication we assume the Riemann integrability of $f\alpha'$ with integral $I$ and start with a arbitrary Riemann-Stieltjes sum $S(P,f,\alpha)$ and a corresponding Riemann sum $S(P, f\alpha') $ and repeat the above argument starting with $|S(P, f, \alpha) - I|$ and prove that $f\in\mathcal{R} (\alpha) $ with integral $I$.

The above argument can be provided using the definition via refinement of partitions by making appropriate changes. Further note that Riemann integrability (and value of integral) of $f\alpha'$ does not change via change of definitions and thus existence of $\int_a^b f\alpha'$ implies the existence of $\int_a^b f\, d\alpha$ under both definitions with same value of integral (this was mentioned in one of the comments to the question).