Generalizing simple functions, let us say we have countable linear combinations of indicator functions over $\mathbb{R}^n$, that is $f(x)=\sum_{i=1}^{\infty}a_i\chi_{E_i}(x)$, where $\{E_i\}$ is a set of countable measurable sets and $a_i$ are scalars.
Then, will such functions have a canonical form, like the simple functions, that is, can they be represented as $f(x)=\sum_ib_i\chi_{G_i}(x)$ for constants $b_i$ and pairwise disjoint measurable sets $G_i$. Also, do these functions form a vector space over $\mathbb{R}^n$. The problem seems quite immediate, but I think the real problem here is convergence, as the summands are infinite. Any hints? How to proceed? Thanks beforehand.
The set isn't well defined. Take $\bigcap E_i\ne\emptyset$, $a_i = 1$ for all $i$.