Using the Axiom of Choice with an infinite Cartesian product I don't know of any definition of the Cartesian product other than something like $$A_{i_1} \times A_{i_2} \times \cdots \times A_{i_n} = \{(a_{i_1}, \dots, a_{i_n}): a_{i_1} \in A_{i_1}, \dots, a_{i_n} \in A_{i_n}\}\text{.}$$
Consider an equivalence relation on $[0, 1]$ defined by $x \sim y \Longleftrightarrow x - y \in \mathbb{Q}$.
This partitions $[0, 1]$ into equivalence classes, from a well-known theorem. 
This book (Rosenthal, A First Look at Rigorous Probability Theory) then states:

Let $H$ be a subset of $[0, 1]$ consisting of precisely one element from each equivalence class (such $H$ must exist by the Axiom of Choice, see page 200).

The Axiom of Choice on p. 200 states:

[G]iven a collection $\{A_{\alpha}\}_{\alpha \in I}$ of non-empty sets..., their Cartesian product $\prod_{\alpha}A_{\alpha}$ is also non-empty.

The definition for Cartesian product in this book is like the one I typed above.
I'm trying to understand what is going on with the Axiom of Choice. Let $a \in [0, 1]$ and consider the equivalence class of $a$, denoted $[a]$. This is non-empty, since $a \sim a$. 
By the Axiom of Choice, $$\prod_{a \in [0, 1]}[a]$$
is non-empty.
How does this make any sense? There are uncountably many $a \in [0, 1]$, so I don't see how we can form an "uncountable Cartesian product."
 A: I'm not sure, but I understand an Element $f\in \prod_{a\in[0,1]} [a]$ as being a function $[0,1]\rightarrow [0,1]$ with $f(a)\in[a]$ for all $a$. One can interpret the vector $(a_1,\dots,a_n)\in A_1\times\cdots\times A_n$ as an analogous function $i\in\{1,\dots,n\} \mapsto a_i \in A_i$.
A: The Axiom of Choice provides, given $any\  set\    I\  $and $ any\  set\  \left \{ A_i \right \}_{i\in I}$ functions $f:I\to \bigcup _{i\in I}A_i$ with $f(i)=a_i\in A_i$. In fact, the Cartesian product, at least in Categorical terms,$\ is$ the set of all such functions, which are conveniently represented as "tuples" $\ (a_i)_{i\in I}\ $of elements in the range of the $f's$. 
A: There are a number of statements which are equivalent to the axiom of choice, i.e. The statement is true if and only if one accepts AOC as true.
Among them is the statement that the Cartesian product of non-empty sets is non-empty. But ignore this for your purposes and look at the AOC itself.
AOC: given a family of non-empty sets one can create a set containing one element from each set in the family.
This then gives the justification for defining the set H containing one (representative) element from each equivalence class.
(Among other statements equivalent to AOC are: Zorn's lemma, cardinal comparability, well ordering principle). 
