In Principles of Mathematical Analysis Chapter 13, Rudin define following things:
An interval in $R^p$ is the set of points x=($x_1,x_2,....,x_p$) such that $a_i \leq x_i \leq b_i$ or the set of points which is characterized by $a_i \leq x_i \leq b_i$ with any or all of the $\leq$ signs repleaced by $<$.Empty set is included and $a_i$ could = $b_i$.
If $A$ is the union of a finite number of intervals, $A$ is said to be an elementary set.
If $I$ is an interval, we define $m(I)=\prod_{i=1}^{p}(b_i-a_i)$ no matter whether is included or excluded in any of the inequalities of $a_i \leq x_i \leq b_i$.
If $A= I_1 \cup I_2...\cup I_n$ and if these intervals are pairwise disjoint, we set $m(A)=m(I_1)+...m(I_n)$. We let $\xi$ denote the family of all elementary set of $R^p$
Rudin states two non trivial properties and let readers to verify them.
(A) if $A \in \xi$, then $A$ is the union of a finite number of disjoint intervals. (I have verified this via using induction)
(B)if $A \in \xi$, the $m(A)$ is well defined by $m(A)=m(I_1)+...m(I_n)$; which means, if two different decompositions of $A$ into disjoint intervals are used, each gives rise to the same value of $m(A)$
I have no idea about how to verify (B) If {$I_1,...I_n)$} and {$I'_1,...I'_n)$} are two different decompositions of $A$. How can I prove that there measure sum are equal to each other? What's the relationship with these two sets?