I'm trying to prove that given the conditions of the Riesz–Markov–Kakutani representation theorem and additionally assuming the space is $\sigma$-compact, the result measure is inner regular for every measurable set.
I am having trouble using the $\sigma$-compactness, since even if I take a compact set from every subspace, a countable union doesn't have to be compact. I also tried taking an open set and then its union, and then its compliment - but it's only closed, doesn't have to be compact. Furthermore, since the measure may be infinite, I don't see a useful connection between a set's measure and it's compliment's measure.
Please only provide a hint, and not a solution.