Why are signed and complex measures typically not allowed to assume infinite values? 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?

It has to do with adding measures to satisfy countable summability.  Notice that in the complex plane $+\infty$ and $-\infty$ are the same value.  So we can't allow one without the other, and who knows what it should mean to add them?  On the other hand if we stick to finite values then the summability condition is clear.
