Anyone know more about reimann-stieltjes integrable functions? I'm researching them and i need to show if is integrable with on continuous, then
Look at each summand defining : it is of the form , where is a subinterval of and .
Well, now apply the mean value theorem to the function , substitute this in every summand of
the sum defining and get the Riemann sum of exactly what you need since we're given is Riemann integrable...
Well, f is cont. and thus Riemann integ., and we're given g' is (Riemann) int., so fg' is Riemann int., and my first post proves the Riemann-Stieltjes int. on the left equals the Riemann int. on the right, so from the existence of the RHS and the equality between both sides it follows the existence of the LHS.
I think my argument is sound, and there's no approximation in it: it shows that the sum whose limit defines the Riemann-Stieltjes integral on the LHS equals the sum whose limit limit defines the Riemann integral on the RHS, and when we pass to the limit in both sides (when the number of partition points goes to infinity and the partition's length goes to zero), this last sum's limit exists since fg' is a Riemann integrable function, as already said, from which also follows the existence of the LHS's linit.
The bounded variation condition on g is fine to know beforehand that the LHS integral exists (together with f being continuous or simply Riemann integrable), but I think we can also deduce its existence by the above argument.