Understanding Integral Notation: Lebesgue-Stieltjes Integral

[oswald2323]

Reading parts of the Monte Carlo methods book by G. Fishman I stumbled into this:

it says that whenever $$\zeta(R) = \int_R \varphi\ dx$$ exists, its value is the same as the Lebesgue-Stieltjes integral $$\zeta = \int_Z \kappa(z)dF(z)$$.

I am confused as to what this means, especially the "dF(z)" part.

Thanks in advance.

 

[mathman]

You need to get a tutorial on Stieltjes integral. Try Google - the Wikipeida articles (Riemann-S and Lebesgue-S, the latter is what is used in the question you raised) should give you a good idea. As far as the Monte Carlo method is concerned, the statement you quoted is a basic theorem of probability theory.