Starting with an additive property for distributions of two statistically independent random variates in terms of different sum functions, we have characterized two general measures associated with two distributions of a discrete random variate. One of these measures is logarithmic, while the other contains powers of variables. An interesting aspect is that under suitable additional boundary conditions the logarithmic measure leads to measure of information (directed divergence) studied by Kullback and measure of inaccuracy studied by Kerridge, while the other solution leads to their parametric generalizations.