TITLE: Measures which cannot be uniquely written as the sum of a purely atomic measure and a nonatomic measure
QUESTION [5 upvotes]: Maharam's theorem says that every complete measure can be written as the sum of a purely atomic measure and a nonatomic measure. According to the paper "Atomic and Nonatomic Measures" by R.A. Johnson, this decomposition becomes unique if we require the measure to be $\sigma$-finite. However, Johnson claims that uniqueness may not hold if we drop the $\sigma$-finite hypothesis. Is this obvious? If not, would someone please direct me to a counterexample?

REPLY [3 votes]: From Example 2.9 in the paper: "Let $\mu$  and $\nu$ be counting measure and Lebesgue measure, respectively." 
You can decompose $\mu$ as $\mu+0$ or $\mu+\nu$ or $\mu+2\nu$, etc... they are all the same measure.