What is your definition of Riemann-Stieltjes Integral?

Question

I'm studying Rudin-PMA and the definition here is different from that of Wikipedia.

Rudin PMA page 122

Wikipedia

Rudin says the condition that "integrator should be monotonically increasing and integrand should be bounded on $[a,b]$" is necessary to define Stieltjes Integral.

But wikipedia defined Stieltjes integral, not assuming above condition.

I'm confused now.. Which definition should i use?

EDIT:

Let $g:[a,b]\to \mathbb{R}$ and $f:[a,b]\to \mathbb{R}$ be functions.

Suppose $\forall \epsilon>0, \exists \delta>0$ such that $\operatorname{mesh}(P)<\delta \implies |S(P,f,g)-A|<\epsilon$.

Then is $f$ bounded and $\inf U(P,f,g) = \sup L(P,f,g)=A$? (where $U(P,f,g)=\sum_{i=1}^n M_i |g(x_i) - g(x_{i-1})|$).

Answer 1

Consider the following slightly more explicit definition:

Suppose $f$ and $\alpha$ are bounded real-valued functions on $[a,b]$ and we have a partition $P=\{x_0,\dotsc,x_n\}$ of $[a,b]$ and set of tags $T = \{t_1,\dotsc,t_n\}$ with $t_i \in [x_{i-1},x_i]$. Then we call

$\qquad \displaystyle S_\alpha(f,P,T) := \sum_{i=1}^n f(t_i) \bigl[\alpha(x_i)-\alpha(x_{i-1})\bigr]$

a Riemann–Stieltjes sum for $f$ with respect to $\alpha$.

We say that $f$ is Riemann–Stieltjes integrable with respect to $\alpha$ and write $f \in \mathcal R_\alpha[a,b]$ when there exists a number $I \in \mathbb R$ such that for every $\epsilon > 0$ there is a partition $P_\epsilon$ such for for all finer partitions $P \supseteq P_\epsilon$ and all choices of tags $T$ compatible with $P$:

$\qquad \displaystyle |S_\alpha(f,P,T)-I|<\epsilon$

and in that case we write $\int_a^b f\,d\alpha = I$; or $(RS)\int_a^b f\,d\alpha=I$ if we want to emphasize that we're referring to the Riemann–Stieltjes integral.

You can easily convince yourself that the above is essentially a restatement of the definition from the Wikipedia page. It's also a slight rephrasing of a passage from Neal L. Carothers - Real Analysis, Cambridge University Press, 2000. Google books (incomplete), Amazon.

In that section, "Integrators of Bounded Variation", he also remarks that for $\alpha$ increasing we have for all partitions that $L(f,P,\alpha) \leq S_\alpha(f,P,T) \leq U(f,P,\alpha)$. In particular it's easy to show that the integral against an increasing function $\alpha$ defined using upper and lower sums (also called the Darboux–Stieltjes integral, written $(DS)\int_a^b f\,d\alpha$ if one wants to be explicit) agrees with the Riemann–Stieltjes integral.

But upper and lower sums don't necessarily make sense when $\alpha$ is only assumed to be of bounded variation. However Carothers has as Exercise 39 in Chapter 14 to prove that one can compute the Riemann–Stieltjes integral $\int_a^b f\,d\alpha$ using the Jordan decomposition $\alpha = p-n$ as the difference of two Darboux–Stieltjes integrals, i.e.

$\qquad\displaystyle (RS)\!\int_a^b f\,d\alpha = (RS)\!\int_a^b f\,dp - (RS)\!\int_a^b f\,dn = (DS)\!\int_a^b f\,dp - (DS)\!\int_a^b f\,dn$,

where the first equality is just linearity in the integrator of the Riemann–Stieltjes integral and the second equality is the one that requires a bit of work.

Answer 2

I don't think Rudin says the conditions are necessary in order to define the integral (although I can't check it right now). Monotonicity is of the integrator and boundedness of the integrand is sufficient, but you can also get away with the integrator being of bounded variation (and bounded integrand) or with both functions being Hölder continuous and the sum of the corresponding Hölder exponents being greater than 1.