# 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!

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## 1 Answer

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

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.

3. 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.)

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.

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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
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