Let $I$ be a special rectangle in $\mathbb{R}^n$, and denote $\lambda(A)$ the measure of $A$. Prove that the following conditions are equivalent:
a) $\lambda(I)=0$
b) $I^{\circ}=\emptyset$ (i.e., the interior of $I$ is empty)
c) $I$ is contained in an affine subspace of $\mathbb{R}^n$ having dimension smaller than $n$. (An affine subspace is any set of the form $\{x_0+x|x\in E\}$, where $x_0\in\mathbb{R}^n$ is fixed and $E$ is a subspace of the vector space $\mathbb{R}^n$. Its dimension is equal to the dimension of $E$.)
The implication $a\implies b$ isn't too bad (using the definition if $I=[a_1,b_1]\times...\times[a_n,b_n]$ then $\lambda(I)=(b_1-a_2)...(b_n-a_n)$. So we conclude $a_i=b_i$ for some $i$. And since $I^{\circ}$ is open, there can't be anything contained in it). I also see how c) makes sense (at least intuitively), but not sure how to show it formally. Thank you.