The perfect set property theorem states that every uncountable Borel set contains a compact subset homeomorphic to Cantor set.
Now suppose that $\mu$ is a regular Borel measure (on some measurable space) such that each set of cardinality smaller than continuum is of measure zero. By the definition of regularity, every Borel set of positive measure has a compact subset of positive measure.
Is it possible to somehow combine the two results above by proving that every Borel set of positive measure contains a compact subset of positive measure which is homeomorphic to Cantor set?