Volume of open box and of closed box In the book Analysis 2, Terence Tao defined the notion of an open box and its volume:

Definition 7.2.1 (Open box)
An open box (or box for short) $B$ in $\mathbb{R^n}$ is any set of the form:
$B = \prod_{i: 1 \to n} (a_i,b_i) := $ {$(x_1,..,x_n)\in \mathbb{R^n}: x_i \in (a_i,b_i) \, for \, all \, 1 <= i < = n$} ($a_i, b_i$ are real numbers).


We define the volume $vol(B)$ of this box to be the number:
$vol(B) := \prod_{i = 1 \to n} (b_i - a_i) =  \, (b_1 - a_1)(b_2 - a_2)...(b_n - a_n)$

By this definition, I understood that the notion of $volume$ is only defined for an open box. The author does not say anything about the volume of a closed box, or a half-closed box.
How can I rigorously extend the notion of a volume of an open box to the notion of a volume of a:

*

*closed box
i.e. $B = \prod_{i: 1 \to n} [a_i,b_i] := $ {$(x_1,..,x_n)\in \mathbb{R^n}: x_i \in [a_i,b_i] \, for \, all \, 1 <= i < = n$}

*half-closed box
$i.e.$ $B = \prod_{i: 1 \to n} (a_i,b_i] := $ {$(x_1,..,x_n)\in \mathbb{R^n}: x_i \in (a_i,b_i] \, for \, all \, 1 <= i < = n$}

EDIT:
What is the point of defining the notion of volume of an open box and not other kind of box (closed, half-closed) ? Is it personal preference of the author or there is some motivations behind ?
 A: These formulae are exactly the same for the closed and half-closed boxes.
In my old measure theory course we used $[a_i, b_i)$-sided-boxes (defined on a semi-algebra and then extended via standard theorems).. It all comes down to the same here.
Usually (I don't know Tao's approach) we define a first approximation of the measure on a small subcollection of all the sets that we eventually want to define it on (here at least the Borel sets) and then apply theory to extend it to that larger collection. Probably Tao can prove the formulae for the other kinds of boxes later, after proving the desired extension exists and is unique, and then he doesn't need to define it here. But I know that these formulae will be the correct ones in the end (and you do too, as you've learnt about volume in primary school, right?)
The starting type of boxes is a matter of preference as to what way of extending Tao will choose. He will have his reasons, that hopefully become clear later in the text.
A: For the final question: you just pick your favorite type of box as your starting point. For this type of box we first define the "usual naive" notion of volume. Then, the idea is that once you properly define the Lebesgue measure (for example by first defining the Lebesgue outer measure using these boxes and then using Caratheodory's procedure to restrict to an appropriate $\sigma$-algebra), you can then prove that for any type of box, the Lebesgue measure of the box equals the product of its side lengths. So ultimately it doesn't matter which way you start.
Although here are just some small details to take note of:

*

*Boxes with one side open one side closed, like $[a_1,b_1)\times \cdots \times [a_n,b_n)$ are nice because if you take two "neighbouring" boxes, then they will be disjoint (for example, $[0,1)$ and $[1,2)$ are disjoint) so maybe in the beginning stages of developing the theory this may come in handy. Another nice thing about such boxes is that any open set in $\Bbb{R}^n$ can be written as a countable disjoint union of such boxes.


*Sometimes, closed boxes may be nicer to work with because they're compact. THis is a crucial fact which is used in the proof I know for the fact that the Lebesgue (outer) measure of any type of box (closed/open/mixed etc) is simply the product of side-lengths. More precisely, $B$ is an open box and $\overline{B}$ is the corresponding closed box, then one can show that for any subset $B\subset S\subset \overline{B}$, we have the outer measure satisfying $m^*(S)=\text{"usual volume" of $B$}$.
