Smith-Volterra- Cantor set. How do I show that the characteristic function $\chi$ of SVC is not Riemann integrable on $[0,1]$ with no using of measure. 
I'd like to show: $U(\chi,P)-L(\chi, P)\geq \frac{1}{2}$
Thanks a lot!
 A: Looks like this has been mostly sorted out in comments. First observation is that the lower sum is always $0$, since the set  does not contain any interval. Let's call the set $F$, and the $n$th step of its construction $F_n$.
It is also easy to show that $\inf U\le 1/2$, using $F_n$ to make suitable $P$. 

The hard part is to   show  $\inf U(\chi,P)\ge 1/2$. It's not enough to look at just one $P$. The estimate must be proved for a general $P$.  
Take the union of all subintervals $I_j$ from $P$ that intersect the set $F$. We must show that the sum of lengths of these subintervals is at least $1/2$. 
To minimize the amount of measure theory involved, you may want to prove that there exists $n$ such that $F_n\subset \bigcup U_j$. This leads to the desired estimate at once, since the $F_n$  is just the union of finitely many intervals whose lengths we know. 
A hint for the existence of $n$ as above: let $\delta$ be the smallest length of a partition interval from $P$ that does not intersect $F$. Choose $n$   large enough, all gaps created at the steps $n+1$, $n+2$,... are shorter than $\delta$. Conclude that any interval from $P$ that does not intersect $F$ also does not intersect $F_n$.
