A question about signed measure?

Question

I have question about the definition of sigma-additive of signed measure.

That is: $\nu(\bigcup E_i) = \sum_{i=1}^{\infty} \nu(E_i)$.

If $\nu(\bigcup E_i)$ is finite,the right-hand side must be absolutely convergence,since if it's not,we can use Riemann Rearrangement Theorem to prove it can converge to any real number.

What I don't understand is why it's not necessary to absolutely converge if $\nu(\bigcup E_i) = \infty$

It's easy to list some examples,I need to understand this question. What is the difference if left-hand side is finite and left-hand side is infinite

Answer 1

If $\nu(\bigcup E_i) = +\infty$ then a fortiori $\sum_{i=1}^{\infty} \nu(E_i) = +\infty$ and so, in particular, $\sum_{i=1}^{\infty}|\nu(E_i)| = +\infty$.

If you chose $-\infty$ instead, you can repeat the same argument with a minus sign in front of the series.