In a number of real analysis texts (I am thinking of Folland in particular), three different kinds of measures are defined.
- Positive measures: Take values in $[0, +\infty]$
- Signed measures: Take values in either $(-\infty, \infty]$ or $[-\infty, \infty)$, but cannot assume both $+\infty$ and $-\infty$.
- Complex measures: Take values in $\mathbb{C}$. Any kind of infinity is not allowed.
My question is: why is this? Is this because of how we set up integrals with respect to these measures, or does it have to do with adding and subtracting measures to make new ones?