Let $E$ and $F$ be Jordan regions. Prove that Vol($E$)$>0$ if and only if $E^\circ \neq \emptyset$. 
Let $E$ and $F$ be Jordan regions.
(a) Prove that Vol($E$)$>0$ if and only if $E^\circ \neq \emptyset$.
(b) Prove that $E$ \ $F$ is a Jordan region.

The above are the problem's I'm stuck on.  For (a), I figured out the part where $E^\circ \neq \emptyset$ implies Vol($E$)$>0$, but I can't seem to figure out how to prove it the other way around.  I'm assuming it involves the definition of volume and finding the smallest possible Jordan region (an open ball of radius $\epsilon$ maybe?) but I can't seem to figure that out.  Also, for (b), I'm unsure on how to approach this.  Would this have two boundary sets (thinking of the set as like a donut or some like shape)?  Or how would it work?
 A: There are many ways to prove that $vol(E) > 0 \implies E^o \neq \emptyset$ depending on which (equivalent) definition of Jordan region you are familiar with.
For one, we say that $E$ is a Jordan region if the boundary $\partial E$ has zero content.  In this case, the indicator function $\chi_E$ is Riemann integrable over an enclosing rectangle $Q \supset E$ -- where the value of the integral is independent of the choice for $Q$ and
$$vol(E) := \int_Q \chi_E$$
We are given that $vol(E) > 0$. Since $\chi_E$ Riemann integrable, it follows that 
$$0 < vol(E) = \int_Q \chi_E = \sup _P L(P,\chi_E),$$
where $L(P,\chi_E)$ denotes the lower Darboux sum with respect to a partition $P$ of  $Q$ into subrectangles.  Hence, there exists a partition $P = \{R_1,R_2, \ldots , R_n\}$ such that
$$0 < vol(E)/2 < L(P,\chi_E) =\sum_{j=1}^n \inf_{x \in R_j} \chi_E(x) vol(R_j)$$
If $R_j \not\subset E$ then there is a point in $R_j$ where $\chi_E(x) = 0$ and $ \inf_{x \in R_j}\chi_E(x) = 0$. On the other hand, if $R_j \subset E$ then $\chi_E(x) = 1$ for all  $x\in R_j$ and $ \inf_{x \in R_j}\chi_E(x) = 1$.
Thus,
$$0 < \sum_{j=1}^n \inf_{x \in R_j} \chi_E(x) vol(R_j) = \sum_{R_j \subset E}vol(R_j),$$
and it follows that there exists at least one rectangle $R_j \subset E$ where $vol(R_j)> 0$, proving that $E_0 \neq \emptyset$.
Another (equivalent) definition is that $E$ is a Jordan region (or Jordan measurable) when the inner and outer Jordan measures are equal, that is
$$|E|_* = \sup_{Q \subset E} vol(Q) = \inf_{Q \supset E} vol(Q) = |E|^*,$$
where $Q$ denotes an elementary set, which is a finite union of non-overlapping rectangles.  Starting with this definition, part (b) is proved here.
