A Characteristic Property of Atomic Measures with Finitely Many Atoms Resp. Atomless Measures

It is proved that a finite measure on a σ-algebra is atomic with a finite number of atoms if and only if there does not exist a {0,1}-valued pure charge on the same σ-algebra whose system of zero sets is larger than the family of zero sets of the finite measure. Furthermore, an example is given which shows that there exists a {0,1}-valued pure charge whose system of zero sets is not larger than the family of zero sets of any finite measure. Finally it is proved that a finite measure on a σ-algebra is atomless if and only if the support of the regular Borel measure of the corresponding Stone representation does not contain a σ-additive measure.