Let $(X,\Sigma)$ a measurable space.
A function $\mu:\Sigma\to\mathbb R$ is called finite signed measure if it is $\sigma -$ additive.
A function $\lambda: \Sigma\to [-\infty,\infty] $ is called signed measure if
i) $\lambda(\emptyset)=0$
ii) $\lambda(\bigcup_{n=1}^\infty A_n)=\sum_{n=1}^\infty \lambda(A_n)$ where $A_1,A_2,\dots$ are disjoint sets in $\Sigma$.
iii) $\lambda$ may take only $\infty$ or $-\infty$ as values, but not both.
The definition I learned is the second. But in a lot of text books about measure theory signed measures are defined like the first definition. Why do they do so? I think proofs will be easier, but don't you lose a lot generality or is there a way to extend results from finite signed measures to signed measures?