Proving Bernstein Set is µ Nullset: A Case Study

[askforhelp]

Hi, i have a question concernign Bernstein sets:
Prove that every µ* measurable set of a Bernstein set is µ nullset.

I know that a set is called Bernstein if both the set and its complement intersect every closed and uncountable set, but i read another definition that says a Bernstein set means there is a compact set set such that its intersection with B and B^c is countable. and fromt his defintion i could apply the theorem that says every measurable set E, its measure is the supremm of the measures of its compact subsets...is this true??

 

[lippyka]

Yes, this is true. To prove that every $\mu^*$-measurable set of a Bernstein set is $\mu$-nullset, we can use the fact that the measure of a $\mu^*$-measurable set is the supremum of the measures of its compact subsets. Since the intersection of a Bernstein set with any closed and uncountable set is countable (or empty), the measure of each compact subset of a Bernstein set is zero. Thus, since the measure of a $\mu^*$-measurable set is the supremum of the measures of its compact subsets, it follows that the measure of any $\mu^*$-measurable set of a Bernstein set is zero. Therefore, we can conclude that every $\mu^*$-measurable set of a Bernstein set is $\mu$-nullset.