Elementary rectangles in product measurable spaces I am working with showing certain equivalence between two probability spaces, one of them being a countable product space of finite spaces, i.e.
$$
  (\Omega,\mathscr F) = \prod_{k=0}^\infty \left(X,2^X\right)
$$
so clearly $\mathscr F$ is generated with measurable rectangles, i.e. the class $\mathscr R$ given by
$$
  \mathscr R:=\left\{\left.\prod_{k=0}^n A_k\times \prod_{k=n+1}^\infty X\right|A_k\in 2^X,n\in \mathbb N_0\right\}.
$$
But I want to use that there is a simpler class that generates $\mathscr F$, namely
$$
  \mathscr R_e:=\left\{\left.\prod_{k=0}^n \{x_k\}\times \prod_{k=n+1}^\infty X\right|x_k\in X,n\in \mathbb N_0\right\}.
$$
which I was going to call "elementary rectangles" - but I am not sure if it is a good name, or if it has been used already somewhere in this or another meaning. What would you advise?
 A: In topology these are often called the basic open sets of $\prod_{k=0}^\infty X$,
because they form a commonly used basis for the product topology on that space.
I personally would write $X^{\mathbb N}$ or $X^\omega$ instead $\prod_{k=0}^\infty X$, though.  But of course, your notation is more flexible as you can easily use it for products that don't have the same space $X$ in every coordinate.
One common notation for your elementary rectangles is $[s]$ where $s=(x_0,\dots,x_k)$.  Here $[s]$ denotes all extensions of the finite sequence (or function, if you wish) in the space $X^{\mathbb N}$.
Not directly related to your question: you are probably aware of the fact that for finite spaces $X_k$, $k\in\mathbb N$, the product $\prod_{k=0}^\infty X_k$
is homeomorphic the Cantor cube $\{0,1\}^{\mathbb N}$ (or $2^\omega$ in more set theoretic notation).  The Cantor cube is homeomorphic to the usual middle thirds 
Cantor set on the real line.
Finally, if $X$ is an uncountable, separable complete metric space (a Polish space), then it is Borel-isomorphic to $\mathbb R$.  I.e., all your spaces together with their Borel $\sigma$-algebras are actually isomorphic (neglecting the case that $X$ could be empty or have only one point) to $\mathbb R$ with its Borel $\sigma$-algebra.
Of course, you can have various different Borel probability measures on $\mathbb R$, some of which are easier to write down if you use another incarnation of the reals, such as $X^\omega$ for some finite space $X$.
