Thanks for reading! My questions are based on the following quotes from Wikipedia:
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?
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.
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!