Problem: Given $f:A\subset \mathbb{R}^n\to\mathbb{R}$ a bounded function, $f(x)\geq 0$. Prove that if for all $\epsilon>0$ exist P partition such that $S(f,P)<\epsilon$ then $f$ is integrable and $\int_A f=0$.
I know that $S(f,P)=\sum_{i_1,...,i_n} f(\xi_{i_1,..,i_n})\mu(R_{i_1,...,i_n}) $ where $\mu(R_{i_1,...,i_n})$ is the "volume" of the n dimensional cube (generated by the partition) and $\xi_{i_1,..,i_n}\in R_{i_1,...,i_n}$. I dont know how to start, maybe via contradiction but i really don't see clearly how to procced.
I will appreciate any help.
This is false for Riemann integrals. Take $A = [0,1]$ and let $f$ be a Dirichlet function (taking the value $1$ at rational points and $0$ at irrational points).
For any partition $P$ we can select irrational intermediate points such that $S(P,f) = 0$. Thus for any $\epsilon > 0$ there is some partition and Riemann sum such that $S(P,f) < \epsilon$. However, $f$ is not Riemann integrable.
I suspect you are missing some other hypotheses.