An independent random cascade measure m is specified by a random generator (w1 ,..., wc), E*Sum w_i = 1 where c is the branching parameter. It is shown under certain restrictions that, if m has two generators with a.s. positive components, and the ratio ln c1/ln c2 for their branching parameters is an irrational number, then m is a Lebesgue measure. In other words, when c is a power of an integer number p and the p is minimal for c, then a cascade measure that has the property of intermittency specifies p uniquely.