TITLE: Is there a maximum to the amount of disjoint non-measurable subsets of the unit interval with full outer measure?
QUESTION [12 upvotes]: This question arose a few years back when I was an assistant teacher on a course of basic (Lebesgue) measure theory, but I didn't find an answer or anyone able to solve the problem. The setting of the problem is as follows:
We say that $A$ has full outer measure in the unit interval $[0,1]$ if 
$$m^\ast(A) = m^\ast([0,1]).$$
How many disjoint subsets of the unit interval with full outer measure can there exist? By the addivity property of the Lebesgue measure these sets have to be non-measurable.
In this MO post Gerald Edgar shows that by using the Bernstein set you can compose any measurable subset $A$ of $\mathbb{R} ^n$ into two disjoint subsets, both with full outer measure in $A$. This gives a partial answer of "at least two" to my question. 

Question: Can the 'Bernstein set -construction' be modified (or is there some other method) to create $n$ disjoint subsets of $[0,1]$ with full outer measure in the unit interval? 
  Can there exist infinitely many disjoint subsets of the unit interval with full outer measure? If so, then can this family be uncountable?

REPLY [13 votes]: In 1917 Luzin and Sierpinski proved there exists continuum many pairwise disjoint subsets of the interval $[0,1]$ such that each of these subsets has outer Lebesgue measure $1.$
Nikolai N. Luzin and Waclaw Sierpinski, Sur une décomposition d'un intervalle en une infnité non dénombrable d'ensembles non mesurables [On a decomposition of an interval into nondenumerably many nonmeasurable sets], Comptes Rendus Académie des Sciences (Paris) 165 (1917), 422-424.
See Gallica site for C. R. Paris volumes or Internet Archive copy of volume 165
Although I've mentioned this paper in various internet groups several times over the past 10+ years, I don't think I've ever mentioned why I find it so fascinating. First, that such an amazingly strong result was published so early -- little more than a decade after non-measurable sets were known to exist. Second, that such a result is so rarely mentioned in texts that deal with measure theory, despite that fact that very little background is needed to state the result, which can also easily be done in a one-sentence footnote.