Herebelow, all passages in Billingsley $(1995)$ to get to definition of product spaces:
The standard construction of the general process involves product spaces. Let $T$ be an arbitrary index set and $R^T$ be the collection of all real functions on $T$. If $T=\{1,2,\ldots,k\}$, [...] $R^T$ can be identified with $k$-dimensional Euclidean space $R^k$. [...] Whatever the set $T$ may be, an element of $R^T$ will be denoted $x$.
So far so good, I guess. As far as I understand, $R^T$ is hust the collection of all the possible maps from $T$ to $R^1$ and, if $T=\{1,2,\ldots,k\}$, $R^T=R^k$. Going on:
Just as $R^k$ can be regarded as the cartesian product of $k$ copies of the real line, $R^T$ can be regarded as a product space - a product of copies of the real line, one copy for each $t$ in $T$
First doubt: as to the immediately-above statement, in plain language (sorry, I am not very comfortable with such concepts by now), what does it mean that $\text{"}R^k\text{ can be regarded as the cartesian product of }k \text{ copies of the real line"}$?
My interpretation: start with $T=\{1,2\ldots,k\}$; corresponding to each $t\in T$, $x_t$ can possibly take every value in the real line $R^1$, hence I would define $R^k$ as follows:
$R^k=\{(x_{1}^{\alpha},...,x_{k}^{\alpha})\hspace{0.3cm}|\hspace{0.3cm}x_{\eta}^{\alpha}\in R^1,\text{ with }\eta\in\{1,\ldots,k\} \text{ and }\alpha\in R^1\}$
Rearranging a bit, I would say that $R^k$ is made up of the cartesian products of all the possible combinations of values of $x_i$ ($i=1,\ldots,k$), with $x_i$ possibly taking values on all the real line $R^1$.
Is that correct? Is there anything wrong in my reasoning?
Second doubt: what does it mean that $\text{"}R^T\text{ can be regarded as a product space - a product of copies of the real line, one copy for each }t \text{ in }T\text{"}$?
My intepretation: I would interpret this statement just as a generalization of the above statement. While in the above statement, starting point is $T=\{1,\ldots,k\}$, now $T$ is generic.
Is that correct? Is there anything wrong in my reasoning?
