Feynman proposed a postulate or a method of quantization in his celebrated paper in 1948. Applying Feynman’s postulate to temporally continuous quantum measurements of the positions of particles, Mensky proposed the restricted Feynman path integrals (RFPIs) for continuous quantum measurements after phenomenological considerations. Our aim in this paper is to give a rigorous proof that Mensky’s RFPIs emerge out of Feynman’s postulate under a simple approximation. In addition, it is proved that the RFPIs emerge out of von Neumann’s postulate on instantaneous measurements as well as Feynman’s postulate. The quantum systems that we study include spin systems. These results are applied to formulations of the multi-split experiments, the quantum Zeno and the Aharonov–Bohm effects.