Decomposition of the Line Into Countably-Many Measure-Theoretic Dense Sets
Joseph Bennish · Missouri Journal of Mathematical Sciences · 1993
Like the warp and woof of a piece of cloth, two sets may be thoroughly intermingled.But how intermingled can disjoint sets be?With this in mind we ask the following question: Can R n be decomposed into countably-many (or even just two) disjoint Lebesgue measurable sets such that the intersection of any one of these sets with any continuous (non-constant) curve has positive one-dimensional Hausdorff measure (or, at least, positive Hausdorff dimension)?(For the definition of Hausdorff