This paper analyses the question of how to redistribute individual incomes if those incomes are determined by the agents' characteristics. It is assumed that some of these characteristics are to be considered relevant , whereas others should be irrelevant . Aredistribution mechanism assigns a post-tax income distribution to each characteristics profile. The objective is to eliminate the influence of irrelevant characteristics while preserving the contributions of relevant characteristics to the individual pre-tax incomes. An axiomatic approach is used to provide characterizations of a specific redistribution mechanism. Furthermore, it is shown that the income function (the function linking pre-tax incomes to characteristics) must have a separability property if some plausible conditions are imposed on a redistribution mechanism.