Riemann integrable proof and notation

Question

For Riemann integrable proof, I see $f \in \Re(\alpha)$. Also I see $U(p,f,\alpha)$.
What does $\alpha$ stand for?
Also to prove Riemann integrability, what do I do at very first step?
I know my questions sound stupid but I do need help to understand the materials.

Answer 1

$\alpha$ is used to denote the density function in a Riemann-Stjeltjes integral. The Riemann-Stjeltjes integral is a generalization of the ordinary Riemann Integral. Given a function $\alpha$, we sum $f(x_i) (\alpha(x_{i+1}) - \alpha(x_i))$ over our partition instead of $f(x_i)$.

To prove Riemann Integrability, start with given $\epsilon > 0$. Then prove that there exists $\delta$ so that for any two partitions with mesh size less than $\delta$, the difference between the riemann (-Stjeltjes) sums is less than $\epsilon$. (This is Cauchy's criterion). One can also work out what the integral is and show that the riemann sums converge to that value.