What is the minimal cardinality of a family of sets of real numbers, each of positive Lebesgue measure, such that every set of real numbers of positive Lebesgue measure contains some member of the family as a subset?


One may work in the Cantor space if desired, and chose compact sets as elements of the family. In particular, the answer is not above the continuum. If this happens to be a known combinatorial cardinal characteristic of the continuum, that would be an acceptable answer, of course.



What is the cofinality of the positive measure sets of reals?
It is equal to cofinality of the null ideal which is also equal to the density of the measure algebra. See theorem 3.2.2 in Bartozynski Judah book.


