It is shown that the lines of action between the photon pairs resulting from a positronium decay are not necessarily in line with the original positronium atom. It is also shown why this «source-independent» effect is not normally observed, although it is observable in principle. Moreover we discuss our initial concerns and some conclusions as they bear on the theory of measurement in quantum mechanics. Source-independence is shown to give a satisfactory response to a special form of the Einstein-Podolsky-Rosen paradox involving pairs of position measurements. It also leads to a nonlocal relationship between position measurements that depends on the width of the position detectors.