Supermeasured: Violating Bell-Statistical Independence without violating physical statistical independence

Abstract: Bell's theorem is often said to imply that quantum mechanics violates local causality, and that local causality cannot be restored with a hidden-variables theory. This however is only correct if the hidden-variables theory fulfils an assumption called Statistical Independence. Violations of Statistical Independence are commonly interpreted as correlations between the measurement settings and the hidden variables (which determine the measurement outcomes). Such correlations have been discarded as fine-tuning'' or aconspiracy''. We here point out that the common interpretation is at best physically ambiguous and at worst incorrect. The problem with the common interpretation is that Statistical Independence might be violated because of a non-trivial measure in state space, a possibility we propose to call ``supermeasured''. We use Invariant Set Theory as an example of a supermeasured theory that violates the Statistical Independence assumption in Bell's theorem without requiring correlations between hidden variables and measurement settings (physical statistical independence).