Exploring Proposition 6.1.2 from D&K's Multidimensional Real Analysis II (Integration)

[Math Amateur]

I am reading Multidimensional Real Analysis II (Integration) by J.J. Duistermaat and J.A.C. Kolk ... and am focused on Chapter 6: Integration ...

I need some help with the proof of Proposition 6.1.2 ... and for this post I will focus on the first auxiliary result ... see (i) ... at the start of the proof ...Near the start of the proof of Proposition 6.1.2 D&K state that :

" ... ... Because b_j - a_j = (b_j - t_j) + (t_j - a_j), it follows straight away that :

\(\displaystyle \text{ vol}_n (B) = \text{ vol}_n (B') + \text{ vol}_n (B'') \)Readers of this post only need to read the very first part of the proof of Proposition 1 (see scanned text below) ... BUT ... I am providing a full text of the proof together with preliminary definitions so readers can get the context and meaning of the overall proof ... but, as I have said, it is not necessary for readers to read any more than the very first few lines of the proof.
Can someone please help me to rigorously prove that \(\displaystyle \text{ vol}_n (B) = \text{ vol}_n (B') + \text{ vol}_n (B'') \) ...Hope someone can help ...

Help will be much appreciated ...

PeterThe proof of Proposition 6.1.2 together with preliminary notes and definitions reads as follows:

Hope that helps,

Peter

 

[Euge]

Hi Peter,

By definition of volume, $$\text{vol}_n(B') = (t_j - a_j)\prod_{k \neq j} (b_k - a_k)\quad \text{and}\quad \text{vol}_n(B'') = (b_j - a_j) \prod_{k \neq j} (b_k - a_k)$$
Hence the sum $$\text{vol}_n(B') + \text{vol}_n(B) = [(t_j - a_j) + (b_j - a_j)] \prod_{k \neq j} (b_k - a_k) = (b_j - a_j)\prod_{k \neq j} (b_k - a_k) = \prod_k (b_k - a_k) = \text{vol}_n(B)$$ as desired.

 

[soloenergy]

Preliminary notes and definitions:

Let B = [a_1, b_1] \times [a_2, b_2] \times ... \times [a_n, b_n] be a closed n-dimensional rectangle in \mathbb{R}^n. We define the volume of B as:

\text{vol}_n (B) = (b_1 - a_1)(b_2 - a_2) ... (b_n - a_n)

Now, let t_j \in [a_j, b_j] for j = 1, 2, ..., n. We define the rectangles B' = [a_1, t_1] \times [a_2, t_2] \times ... \times [a_n, t_n] and B'' = [t_1, b_1] \times [t_2, b_2] \times ... \times [t_n, b_n].

Proposition 6.1.2: Let B, B', B'' be as defined above. Then, \text{ vol}_n (B) = \text{ vol}_n (B') + \text{ vol}_n (B'').