The commonly known properties of continuous measures such as the symmetry o
f the A, condition, its equivalence to the reverse Holder inequality, the l
eft-openness of the A, condition, etc., are no longer necessarily true when
the underlying measure is allowed to have atoms. The measures that preserv
e these properties are called good measures. The class of good measures is
investigated and various criteria for a measure to belong to this class are
presented.