Volume in higher dimensions is well defined

Question

If $A=I_1\times\cdots\times I_n$, where the factors on the cross product are pairwise disjoint real intervals, define the volume of A by $vol (A)=length(I_1)\cdot\ldots\cdot length (I_n)$. On the other hand, if $B=A_1\cup\cdots\cup A_m$, where the terms of the union are pairwise disjoint cross products of real intervals (like as the set A), define the volume of B by $vol(B)=vol(A_1)+\cdots + vol (A_m)$. I’m trying to show that this notion of volume in higher dimensions is well defined, in the sense that if $B=\cup A_i$ and $B=\cup B_i$, then the volumes would be the same. Any ideas?