From what I've read in Wikipedia, $\sigma$-additivity is defined in the following way: Let $\mathcal{A}$ be a $\sigma$-algebra of some underlying set $X$ and let $f$ be a mapping $f:\mathcal{A}\rightarrow M\subseteq\mathbb{R}\cup\left\{ \pm\infty\right\} $. Then $f$ is called $\sigma$*-additive*, if for any countable set $I$ and a family of pairwise disjoint sets $\left(A_{i}\right)_{i\in I}$, we have $ f\left(\bigcup_{i\in I}A_{i}\right)=\sum_{i\in I}f\left(A_{i}\right). $
My questions are: 1) Did we even need the fact that $\mathcal{A}$ is a full-blown $\sigma$-algebra for this definition ?
Besides using the fact that $\mathcal{A}$ being a $\sigma$-algebra guarantees me that $\bigcup_{i\in I}A_{i}$ is also in $\mathcal{A}$, at no point in this definition are we using the other properties of $\mathcal{A}$, so we could just as well define $\sigma$-additivity for $f:\mathcal{A}\rightarrow M\subseteq\mathbb{R}$, where $\mathcal{A}$ is just some system of sets such that $\bigcup_{i\in I}A_{i}\subseteq\mathcal{A}$, for pairwise disjoint sets $A_i$.
A different approach: We could drop the above property of $\mathcal{A}$ altogether , so that $\mathcal{A}$ is just system of sets without any additional prperties, and define $f$ to be $\sigma$-additive only for those $\bigcup_{i\in I}A_{i}$ that are contained in $\mathcal{A}$ .
2) Is it custom for $M$ to be some certain subset of $\mathbb{R}$, or can $M$ be an arbitrary one ? I've seen $M=\mathbb{R}\cup\left\{ \pm\infty\right\} $ (on wikipedia) and $M=\left[0,1\right]$ (when dealing with probabilities), so I'm wondering if in the definition of $\sigma$-additivity it is ok to require just $M\subseteq\mathbb{R}\cup\left\{ \pm\infty\right\} $.