Why does the interior of a subset $A \subset E^n$ have the same volume as the subset? The official Question Statement is the following:
Show that if a subset $A \subset E^n$  has volume, then the interior of A has the same volume. 
My guess is that the boundary can be shown to have zero volume, by showing its volume is $\lt \epsilon $ for any $\epsilon$. Therefore the volume of the interior must equal the volume of the entire subset. Unfortunately I am struggling with turning this into a proper proof, and to be frank I am not even sure that I am taking the correct approach.  

Suppose E is R-Q, the interior of E is empty and the volume of E is infinite.
You can also take E a ball of $R^n$ with its rational point removed. So your statement is not true.
