How to rigourously define $G^{|A|}$? where G is a group and A being an infinite set. It is clear to me that $G^n:=G\times G\times ...\times G$. Every member of $G^n$ looks like $(g_1,g_2,...g_n)$.
But how to rigorously define $G^{|A|}$, i.e how to explicitly write elements of $G^{|A|}$?

I've never seen this before but I guess a reasonable way to define $G^\mathbb{R}$ would be to say that $f\in G^\mathbb{R}$ is a map
$$f:\mathbb{R}\longrightarrow G $$
And if I have $f,g\in G^\mathbb{R}$ then $(fg)[x]=f(x)g(x)$, this is a generalization of the $G^n$ case since you can think about elements of  $G^n$ in a similar way as maps from $\{1,...,n\}\rightarrow G$.
