Ergodic properties of p-adic transformations have been studied with respect to Haar measure. This paper extends the study of these properties to measures beyond Haar measure. Under these measures, coefficients do not appear in equal proportions. Adding a rational number that is not an integer then takes likely strings of coefficients to one of two unlikely strings of coefficients. It follows from this inequality that translation by a rational number other than an integer is singular with respect to these measures. Conjugation gives similar results for multiplication by rational numbers.