Pi product + set builder notation - how to read

Question

From the low-discrepancy wiki I have:

$$\displaystyle \prod_{i=1}^s[a_i, b_i) = \{𝑥 \in \mathbb{R}^s : a_i \le x_i \le b_i\}$$ where $$0 \le a_i < b_i \le 1$$

How can I read it? Should I first make do a cartesian product of $[a_i,b_i)$? For example for $s=3$ I would have: $$ [a_1, b_1) \times [a_2, b_2) \times [a_3, b_3) = \{ \{a_1, a_2, a_3\}, \{a_1, a_2, b_3\}, \{a_1, b_2, a_3\}, \{a_1, b_2, b_3\}, \{b_1, a_2, a_3\}, \{b_1, a_2, b_3\}, \{b_1, b_2, a_3\}, \{b_1, b_2, b_3\}\} $$

then I would put it into set-builder notation.. but for first $\{a_1, a_2, a_3\}$ I don't have $b_i$ value.

Answer 1

You should read, $\displaystyle \prod_{i=1}^s[a_i,b_i)$ as, "the product of all intervals closed $a_i$, open $b_i$ from $i=1$ to $s$". And $\displaystyle \{\tilde x\in \Bbb R^s:\tilde x=(x_1,x_2,\dots x_s) \text{ where }x_i\in[a_i,b_i),\forall i=1,2\dots s\}$ as, "the collection of all $\tilde x$ ($x$ tilde) in $\Bbb R^s$ such that $x$ tilde is of the form of $s$-tuple $(x_1,x_2,\dots x_s)$ where $x_i$ belongs to $[a_i,b_i)$ for all $i=1,2,... s$".

Hope it works.

Further, $\displaystyle \prod_{i=1}^s[a_i,b_i]=[a_1,b_1]\times [a_2,b_2]\times ... \times [a_s,b_s]$ where $\times$ is the cartesian product defined here.