Let $\Omega$ be a countably infinite set. Can one use the $\sum_{x\in\Omega} E$ notation when $\Omega$ is countably infinite? I have only seen this being used when $\Omega$ is a finite set.
To make the question more concrete, suppose I have a function $f\colon\Omega\to\mathbb R_{+}$ in $\ell_1(\Omega)$.
Is it a valid notation / good practice to write \begin{align*} \sum_{x\in\Omega} f(x) \end{align*}
or one should rather let $\mu = \#(\Omega)$ and consider the measure space $(\Omega, \mathcal{P}(\Omega), \mu)$ and denote the sum as \begin{align*} \int_\Omega f\,\mathrm{d}\mu\;\;? \end{align*}
The reason I am suspecting that the first expression is frowned upon is that $\sum_{i=1}^\infty$ is defined as $\lim_{n\to\infty}\sum_{i=1}^n$ however it is not clear how to write a corresponding limit when $\Omega$ is an arbitrary countably infinite set (as opposed to $\Omega=\mathbb N$).
Thanks
It turns out, if $f\in\ell^1(\Omega)$, then it makes sense to write \begin{align*} \sum_{x\in \Omega} f(x) \end{align*} since the order in which we perform the sum does not matter in this case, due unconditional convergence of the underlying series.
If $f\notin\ell^1(\Omega)$, then one has to fix an explicit bijection $b\colon\mathbb{N}\to\Omega$ and explicitly write
\begin{align*} \sum_{i=1}^\infty f(b(i)) := \lim_{n\to\infty}\sum_{i=1}^nf(b(i)) \end{align*}
since in general this sum depends on the chosen bijection. (e.g., https://en.wikipedia.org/wiki/Riemann_series_theorem)
Please let me know if this makes sense.
Thanks