Is this analysis of Stieltjes assimilation deal with?
If $f$ is a favorable function, the instinctive analysis of the Riemann indispensable
$\int_a^b f(x) dx$
is the location under the contour $f$ in between $a$ and also $b$.
Intend $f$ and also $g$ are smooth favorable features. Is it proper to analyze the Riemann - Stieltjes indispensable
$\int_a^b f(x) d g(x)$ as the quantity under a "bow", where the elevation of the bow at a factor $u$ in between $a$ and also $b$ is established by $f$ and also the density is established by $g'$?
You are properly changing $\int_a^b f(x) dg(x)$ with $\int_a^b f(x) g'(x) dx$. A quick trip to Wikipedia discloses that this is flawlessly great when $g$ is definitely continual, or when it has a continual by-product, yet might not stand or else:
If $g$ needs to take place to be almost everywhere differentiable, after that the indispensable might still be various from the Riemann indispensable $$\int_a^b f(x) g'(x) \, dx,$$ as an example, if the by-product is boundless. Yet if the by-product is continual, they will certainly coincide. This problem is additionally pleased if $g$ is the (Lebesgue) indispensable of its by-product ; in this instance $g$ is claimed to be definitely continual.
Nonetheless, $g$ might have dive stoppages, or might have acquired absolutely no virtually anywhere while still being continual and also raising (as an example, $g$ can be the Cantor function), in either of which instances the Riemann-- Stieltjes indispensable is not recorded by any kind of expression entailing by-products of $g$.
Modify : I really did not see words "smooth" in your inquiry. In this light, the response to your inquiry is of course.
Additionally, MathWorld is probably a better reference : both expressions are equivalent "if $f$ is continual and also $g'$ is Riemann integrable over the defined period".