the question goes like this:
Let $ f: \mathbb R^n \rightarrow \mathbb R $ be a function such that for every open ball $ B \subset \mathbb R^n $, f is integrable on $ B $ and satisfies $$ \int_B f = 0 $$ Prove that for every Jordan bounded set $ E \subset \mathbb R^n $ f is integrable on $ E $ and satisfies: $$ \int_E f = 0 $$
- Jordan bounded sets are sets with negligible (zero measure) boundary.
I'm stuck with the second part of the question, proving that $ \int_E f = 0 $. I think the way to prove this is to show that for every $ \epsilon > 0 $ we can find open balls $ B_1 , \ldots, B_N $ such that $\forall 1\le i\le N: B_i \subseteq E $ and $ Vol(E \setminus \cup_{i=1}^{N} B_i) < \epsilon $, but I did not succeed to prove this.
- $ Vol(F) $ is the volume of the set $ F $.
We have already proven in our class that if $ E $ is a Jordan bounded set, then the same proposition holds except that the balls are replaced with compact cuboids, but I also don't know how to use this to prove the proposition with the open balls.
Thanks for the help.