I'm trying to find an explaination of the definition of Riemann-Stieltjes Integral used on page 22 of Edwards book [RZ]:
This can also be accessed hopefully legally here: Zeta
1) The question is where does he get 'the usual definition' of the Stieltjes Integral that I have highlighted? Any references or explaination would be good. I looked at Apostol's Mathematical Analysis, Second Edition but he makes no mention of this definition at least in a way that I can understand.
2) If I understand him correctly for example $\pi(x)$ would be $\pi(n-1)+\frac{\pi(n)-\pi(n-1)}{2}$ in the use of the prime number theory as applied to say finding the sum over primes for example $\int_a^x f(t)d\pi(t)$?
RZ, Riemann's Zeta Function (Dover Books on Mathematics) Paperback – 28 Mar 2003 by H M. Edwards (Author)
The idea is that changing the value of $g$ at an internal point $x_0$ doesn't change the value of $\int_a^b f(x) dg(x)$, provided that $f$ is continuous at $x_0$. The integral sum will contain either a term $$f(\xi) (g(x_0 + \epsilon_2) - g(x_0 - \epsilon_1)) \to f(x_0) (g(x_0 + 0) - g(x_0 - 0))$$ if $x_0$ is not a partition point or a sum of two terms $$f(\xi_1) (g(x_0) - g(x_0 - \epsilon_1)) + f(\xi_2) (g(x_0 + \epsilon_2) - g(x_0)) \to \\ f(x_0) (g(x_0) - g(x_0 - 0)) + f(x_0) (g(x_0 + 0) - g(x_0)) = \\ f(x_0) (g(x_0 + 0) - g(x_0 - 0))$$ otherwise.
If $x_0 = a$ or $x_0 = b$ and $f(x_0) \neq 0$, then two different values of $g(x_0)$ will give two well-defined but different integrals.