In a measure space, let's call a measurable subset atomless wrt the measure, if it does not have an atomic subset. In particular a measurable subset with zero measure is atomless.
There may be measurable subsets that are neither atomic nor atomless, for instance, the union of atomic subset(s) and atomless subset(s) with positive measure(s).
- I was wondering if the converse is true, i.e. if a measurable subset is neither atomic nor atomless, then must it be the union of atomic subset(s) and atomless subset(s) with positive measure(s)?
- I think the previous question is equivalent to whether any measurable subset can be partitioned into atomic subset(s) and atomless subset(s)?
Thanks and regards!