computing $\int_{[a,b]}g^\sigma\mathrm{d}g$ for the increasing & continuous function $g$

Question

I encountered this question in Makarov & Podkorytov, Real analysis.

I need to show that for an increasing and continuous function $g$ defined on $[a,b]$ the following formula holds $$\int_{[a,b]}g^\sigma\mathrm{d}g = \frac{g^{\sigma+1}(b)-g^{\sigma+1}(a)}{\sigma+1}$$ for every $\sigma>0$.

I'm having difficulties proving this. Using the definintion of lebesgue integration one can show that this term is from above and below by similar riemann-stieltjes integrals, but this seems like a messy approach, since i think this question is still very well contained within the theory of lebesgue integration.

Any hints are appreciated, but i am looking for an approach which is more lebesgue-integration oriented.

Answer 1

Hint: integrate by parts: $$ \int_a^b g^\sigma dg=g^{\sigma+1}(b)-g^{\sigma+1}(a)-\int_a^bgd(g^\sigma)= g^{\sigma+1}(b)-g^{\sigma+1}(a)-\sigma\int_a^bg^\sigma dg, $$ and the result follows.