Transforming spatial point processes into Poisson processes

In 1986, Merzbach and Nualart demonstrated a method of transforming a two-parameter point process into a planar Poisson process of unit rate, using random stopping sets. Merzbach and Nualart's theorem applies only to a special class of point processes, since it requires two restrictive conditions: (F4) condition of conditional independence and the convexity of the 1-compensator. (F4) condition was removed in 1990 by Nair, but the convexity condition remained. Here both (F4) condition and the convexity condition are removed by making use of predictable sets rather than stopping sets. As with Nair's theorem, the result extends to point processes in higher dimensions.