In this talk, we present a geometric characterization of Radon measures in Rn whose orthogonal projections onto (n-1)-planes are singular with respect to the (n-1)-dimensional Hausdorff measure, based on the so-called probabilistic injective projection property. Specifically, we prove that if a Radon measure μ is not supported on a single hyperplane, then it projects singularly with respect to H^(n-1) if and only if a typical orthogonal projection is injective on a set of full μ-measure. As a consequence, we obtain a characterization of codimension-one pure H^(n-1)-unrectifiability within the class of measures that disintegrate atomically with respect to orthogonal projections. In particular, if μ admits a non-trivial absolutely continuous component under projection along a positive measure set of directions, then μ must contain a non-trivial rectifiable part. This result can be viewed as a measure-theoretic analogue of the classical Besicovitch–Federer projection theorem, adapted to the setting of Radon measures. Finally, we show how this framework provides new insight into a longstanding conjecture of Marstrand concerning radial projections of purely unrectifiable sets with finite Hausdorff measure.