Lebesgue steiljes outermeasure

[jem05]

Homework Statement

1) does there exist an outer measure where every m* nullset is countable?
2) does there exist a lebesgue steiljes outer measure where every m* nullset is countable?

Homework Equations

The Attempt at a Solution

1) i got an example for it , turned out to be easy and the answer is yes.
2) my intuition is no,
i got some information, like, the set of discontinuities of the corresponding function is countable, and of positive measure.
that every point of continuity is of measure 0 and a point of discontinuity is of +ve measure.
but I am stuck, i cannot proceed.
any ideas? thanks a lot.

 

[Eynstone]

does there exist a lebesgue steiljes outer measure where every m* nullset is countable?

I'm not sure, but the Cantor set is uncountable & has measure zero.