We consider the problem of localizing point sources on an interval from possibly noisy measurements. In the absence of noise, we show that measurements from Chebyshev sys- tems are an injective map for non-negative sparse measures, and therefore non-negativity is sufficient to ensure unique- ness for sparse measures. Moreover, we characterize non- negative solutions from inexact measurements and show that any non-negative solution consistent with the measurements is proportionally close to the solution of the system with ex- act measurements. Our results substantially simplify, extend, and generalize the prior work by De Castro et al. [1] and Schiebinger et al. [2], which relies upon sparsifying penal- ties, by showing that it is the non-negativity constraint, rather than any particular algorithm, that imposes uniqueness of the sparse non-negative measure, and by extending the results to inexact samples.