TITLE: How is Riemann–Stieltjes integral a special case of  Lebesgue–Stieltjes integral?
QUESTION [14 upvotes]: Thanks for reading! My questions are based on the following quotes from Wikipedia:

About the existence of 
Lebesgue–Stieltjes integral:

The Lebesgue–Stieltjes integral $  \int_a^b f(x)\,dg(x)$ is defined
  when
  ƒ : [a,b] → R is Borel-measurable and
  bounded and g : [a,b] → R is of
  bounded variation in [a,b] and
  right-continuous, or when ƒ is
  non-negative and g is monotone and
  right-continuous.

I was wondering if this is the right
condition for its existence?
About the existence of
Riemann–Stieltjes integral:

The best simple existence theorem
  states that if f is continuous and g
  is of bounded variation on [a, b],
  then the integral exists. A function g
  is of bounded variation if and only if
  it is the difference between two
  monotone functions. If g is not of
  bounded variation, then there will be
  continuous functions which cannot be
  integrated with respect to g. In
  general, the integral is not
  well-defined if f and g share any
  points of discontinuity, but this
  sufficient condition is not necessary.
On the other hand, a classical result
  of Young (1936) states that the
  integral is well-defined if f is
  α-Hölder continuous and g is β-Hölder
  continuous with α + β > 1.

For the question in the part 3, I
was wondering for Riemann–Stieltjes
integral $\int_a^b f(x) \, dg(x) $
to exist, must g be nondecreasing?
It looks like not the case quoted
above.
Specialization from Lebesgue–Stieltjes integral to Riemann–Stieltjes integral:

Where f is a continuous real-valued
  function of a real variable and g is a
  non-decreasing real function, the
  Lebesgue–Stieltjes integral is
  equivalent to the Riemann–Stieltjes
  integral,

I was wondering  why it only mentions the case when g is nondecreasing? Is this the necessary condition for existence of Riemann-Stieltjes integral?
Do Lebesgue–Stieltjes integral and
Riemann–Stieltjes integral generally
use the same notation $  \int_a^b f(x)\,dg(x)$? How does one
know which one the notation refers
to?

Thanks for helping!

REPLY [4 votes]: I don't think my understanding is completely correct and should have posted as a comment, but it has length restrictions.

It seems to me that when $g$ is BV and right continuous, and $f$ is Borel measurable, $f$ does not have to be bounded. There are unbounded Lebesgue integrable functions, so the same should be true for Lebesgue-Stieltjes integral, which is just Lebesgue integral w.r.t. the signed measure $\mu_g$ on $\mathcal{B}(\mathbb{R})$ induced by $g$.
It should be OK for function $g$ with bounded variation. We can write $g$ as the difference of two non-decreasing functions.
I don't think Riemann-Stieltjes integral requires the integrator to be non-decreasing. It might be BV or possibly  an even broader class of functions. I guess the author of Wikipedia entry mentions only nondecreasing functions because s/he has CDF of a random variable in mind and wants to discuss its application in probability theory.
I also have the impression that these two integrals agree whenever the Riemann-Stieltjes integral exists. (Just like the relation between the Lebesgue integral and the Riemann integral.)
If these two notions agree, there's no danger of using the same notation. Otherwise, I've seen authors using prefix to distinguish different types of integrals, e.g., $(R)\int_a^b...$ for Riemann(-Stieltjes) integrals.