I don't understand what it means for a set to have zero content. What is its geometric interpretation? Can you give me some examples of sets in Rn that do not have any zero contents? Also, can you clarify the primary relationship between zero content and integration in Rn
A set with zero content is a subset of $\Bbb R^n$ that can be covered by finitely many rectangular $n$-parallelepipeds whose combined size can be chosen as small as you want (the size is the product of the lengths of the sides).
A set with zero content necessarily has Lebesgue measure zero and therefore integration over such sets always outputs zero.
An example of a subset of $\Bbb R^n$ without zero content is for instance $[0,1]^n$. If you cover this set by finitely many rectangular parallelepipeds, their combined size has to be at least $1$.