The distribution function associated with a Borel measure (Folland Theorem 1.16)

Question

Here is Theorem 1.16 page 35 from Folland's Real Analysis.

I don't quite understand the reverse part of the theorem. When $x<0$, what is $\mu((-x,0])$? Since $-x >0$, how do we define the measure of the interval $(a,0]$, where $a>0$? And why the defined function is the distribution function associated with $\mu$?

Answer 1

It's a typo. It should simply be $-\mu((x, 0])$ for $x<0$. I believe it's been corrected in later editions (well, it's corrected on the Kindle edition anyway).