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.