The nature of the Lüders projection and its relationship with the von Neumann projection are clarified. It is shown that Lüders-like transformations can be obtained by local unitary transformations on single and product states, but not entangled states. Several examples are given to illustrate this. The distinction between the two projections is further clarified by the analyses of the measurement process on both entangled and single-particle systems. A possible experimental test of the Lüders postulate for non-degenerate systems is also proposed.