We present five axioms for fairness measures in resource allocation. A family of fairness measures satisfying the axioms is constructed. Special cases of this family include α-fairness, Jain's index, and entropy. Properties of fairness measures satisfying the axioms are proven, including Schurconcavity. Among the engineering implications is a generalized Jain's index that tunes the resolution of fairness measure, a new understanding of α-fair utility functions, and an interpretation of "larger α is more fair". We also construct an alternative set of axioms to capture system efficiency and feasibility constraints.