real analysis - Definition of signed measure (assume at most one values of $pm infty$)

In the real analysis (Folland), the definition of signed measure is following:

I am confused about the second one. My professor said that if there were no second requirement, then countable additivity does not mean anything since we can rearrange the sum to get whatever I want.

I have no idea what he was talking about. Could anyone provide me with a concrete example to say anything about this?

Answer

As ForgotALot said, the issue stems from dealing with unions of a set of infinite positive measure and infinite negative measure. While we can handle things like $3+\infty = \infty$, allowing $-\infty$ at the same time is an issue: is $\infty-\infty = 0$ or maybe it is $(3+ \infty)-\infty = 3$?

Here is a concrete example: for subsets of $\mathbb Z$, define $\nu(A)=\sum_{a\in A} \operatorname{sign} (a)$. Since
$$
\mathbb{Z} = \{0\} \cup\bigcup_{n=1}^\infty \{n,-n\}
$$
where each set on the right has zero measure, it follows that $\nu(\mathbb{Z})=0$. But we can also write
$$
\mathbb{Z} = \{0 , 1 \} \cup\bigcup_{n=1}^\infty \{n+1,-n\}
$$
and conclude that $\nu(\mathbb{Z})=1$. So we can't have nice things like countable additivity if both $\pm \infty$ are allowed.