# 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:

1. 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?

2. 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.

3. 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?

4. 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!

1. 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 $g$.
2. It should be OK for function $g$ with bounded variation. We can write $g$ as the difference of two non-decreasing functions.
4. 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.
• Thanks! For 2, I was wondering if $g$ being non-decreasing implies $g$ having bounded variation? Do I need to specify both are over an interval $[a,b]$? – Tim Apr 10 '11 at 21:43
• (1) $g$ nondecreasing implies it has bounded variation, because the total variation of $g$ on $[a,b]$ is always $g(b)-g(a)$ regardless of the partition. In fact, "A function $g$ is of bounded variation if and only if it is the difference between two monotone functions." (2) Yes, we are talking about functions on $[a,b]$. – GWu Apr 10 '11 at 22:24