proving that the graph of a function is of Jordan measure zero Let $f$ be an integrable function from $B$ to $[0,\inf]$ where $B$ is a sphere in $\mathbb{R^n}$.
Exercise: For $f$ and $B$, the graph
$$
\Gamma=\{(x,f(x)):x\in B\} \subset \mathbb{R}^{n+1}
$$
is of volume zero. Prove it.
I am having a hard time proving and understanding the whole notion of when a group is of Jordan measure zero and when it isn't; on the same note, what is the intuition behind proving that a group is Jordan Measurable? For instance, what about a sphere in $\mathbb{R^n}$? Thanks a lot!
 A: By definition, if $R\,$ is a rectangle containing $B$ and $\hat f$ is the zero-extension of $f$ to $R$, then, for every $\varepsilon>0$, there exists a partition of $R$ in subrectangles $S$ with $$\sum_S\, [M_S(\hat f)-m_S(\hat f)] \cdot v(S)<\varepsilon$$ Note that the first member of the inequality is the sum of the volumes of the cartesian products $$[m_S(\hat f),M_S(\hat f)] \times S$$ which are rectangles in $\mathbb{R^{n+1}}$.
So the graph of $f$ has content zero since it is included in the union of those rectangles.
As to your question about proving the Jordan measurability of a (euclidean) n-ball, I will use the theorem:
If $B \subset \mathbb R^n$ is a compact set whose boundary has content zero and $f$ is a continuous function on $B$, then $f$ is integrable on $B$ (and so the graph of $f$ has content zero).
I proceed by induction on n to prove that the boundary of the unit n-ball has content zero.
The boundary of the unit 1-ball $[-1,1]$ has obviously content zero.
Now assume that the boundary of the unit n-ball has content zero. The function $f$ defined by $$f(\mathbf x)=\sqrt {1-\|\mathbf x\|^2}$$ is continuous on the unit n-ball. Its graph is the boundary of an half of the unit (n+1)-ball and has content zero (see the above theorem). The boundary of the other half (consider the function $-f$) has also content zero. Hence the boundary of the unit (n+1)-ball has content zero being the union of two sets having content zero.
Any n-ball is obtained from the unit n-ball by a scaling or/and a translation that preserve the zero content.
