Let \(H\) be a compact convex set in \(R^{k}\) with nonempty interior. Let \(f \in \mathcal{C}(H)\), put \(f(\mathbf{x})=0\) in the complement of \(H\) and define \(\int_{H} f\)

Prove that \(\int_{H} f\) is independent of the order in which the integrations are carried out.

Hint: Approximate \(f\) by functions that are continuous on \(R^{k}\) and whose supports are in \(H\).