I am following a course on stochastic calculus. This section of the notes briefly defines variation, signed measures and the Lebesgue Stieltjes integral of a function with respect to a function of finite variation. I understand the notes but do not understand how the Example 7.4 is calculated:
How is $\mu([a,b))$ calculated? I know that since sets of the form $[0,t)$ generate the real Borel sigma algebra, it is enough to define a measure on sets of this form. How has the author used the definition $\mu([0,t)) = \mu_+([0,t)) - \mu_-([0,t))$ to obtain the result for $\mu([a,b))$? Also, is the integral $(f \cdot a)(t)$ calculated explicitly using the limit of sums as in the definition, or is there a trick im missing?
Thanks
*Edit: I have found that $\mu([0,t)) = a(t)$. I presume that the author obtains $\mu([a,b))$ by doing $\mu([0,b)) - \mu([0,a)) = a(b) - a(a)$ which gets the correct result (a here is a function and a constant annoyingly). But how come we can do this? I get that $[a,b) = [0,b)/[0,a]$ and so that $\mu([a,b)) = \mu([0,b)/[0,a])$. But why then does this become $\mu([0,b)) - \mu([0,a))$. Or maybe that is completely wrong?