non-Lebesgue-measurable subsets of Smith-Volterra-Cantor sets

Question

For the Smith-Volterra-Cantor set (or simply SVC) we define an equivalence relation R by making each connected component in SVC an equivalence class. It is easy to see that the collection of all R-equivalence classes has the cardinality continuum $\aleph_1$. We know that there is a union of R-equivalence classes that is not Borel.

Is there a union of R-equivalence classes not Lebesgue measurbale?

Answer 1

There are a few remarks to be made here:

1. The cardinality of the continuum is not $\aleph_1$ without assuming the continuum hypothesis.

2. SVC sets are totally disconnected. This means that every equivalence class is in fact a singleton.

3. Every set of positive measure has a subset which is non-measurable. So if the SVC set was not of measure zero, this set would be non-Lebesgue measurable as well.