I'm reading this book Probability & Measure Theory by Ash. I think I've come across a part that is a little hand-wavy. We are trying to build a Lebesgue-Stieltjes measure from a distribution function $F$ (in that the measure of interval $(a,b]$ is $F(b) – F(a)$).
He starts by adding $+\infty$ and $-\infty$ to the real line so that we can work in compact space. He defines right-semiclosed as intervals of the form $(a, b]$ and $[-\infty, b]$ and $(-\infty, b]$. He then constructs a field by taking all finite unions of these right-semiclosed intervals.
He defines a set function over this field defined in the intuitive way (the set function takes $(a,b]$ to $F(b) – F(a)$), and he shows that this set function is countably additive.
This is where I don't understand his argument. He seems to say, ignore these points $+\infty$ and $-\infty$ so that our field no longer uses the compact space, and our set function now becomes a proper measure over a real field. Then apply the Carathéodory Extension Theorem.
I don't see how we can go from a compact space to a non-compact space without causing harm to the properties of our set function. I'm hoping that this construction method is widely used, and someone can explain where I am confused. This is Theorem 1.4.4 in Ash, 2nd Edition.
The complete exposition can be found at http://books.google.com/books?id=TKLl3CGqsTEC&lpg=PP1&dq=probability%20and%20measure%20theory&pg=PA22#v=onepage&q&f=false from the bottom of page 22 to page 24.
Best Answer
There is an equivalence between Stieltjes measures on the real line, and Stieltjes measures on the compactified real line that assign measure 0 to the added boundary points at infinity.
For any measure of one type there is a unique measure of the other type that assigns the same values to finite intervals in $R$. This is nothing more than the notation of "improper integrals" from calculus. As in calculus it is a notational convenience used to avoid constantly writing about limits of integrals. The proof could be written or read in terms that avoid any extension of the space or the measure, just as any calculation with improper integrals can be presented as a limit of calculations on finite intervals.