How is Riemann–Stieltjes integral a special case of  Lebesgue–Stieltjes integral? 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!
 A: 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.
