So I'm having a tough time understanding Riemann-Stieltjes and a few of the definitions. I have a question that I think I know the answer to but would like some verification. The question is:

Let f:[a,b]-R be bounded ans $\alpha$:[a,b]-R be monotonically increasing.

a) For a partition P of [a,b], define the upper and lower R-S sums with respect to $\alpha$.

b)Define what it means for f to be Riemann Stieltjes Integrable with respect to $\alpha$

c) State Riemanns Integrability Criterion

My answers a) U(P,f,$\alpha$)=$\sum\limits_{ i=1}^n Mi \delta \alpha i$ where $\delta$i=xi-x(i-1)

L(P,f,$\alpha$)=$\sum\limits_{ i=1}^n mi \delta \alpha i$

b) infU(P,f,$\alpha$)= $\int_{a}^{b} fd\alpha$

$supL(P,f,\alpha)=\int_{a}^{b} fd\alpha$

So this is integrable with respect to alpha if these two are equal.

c) For every $\epsilon$>0 there exits a partition P of I such that U(P,f,$\alpha$)-L(P,f,$\alpha$)< $\epsilon$

Are these solutions correct?


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