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We construct unbiased estimators for the distribution of the number of points inside random stopping sets based on a Poisson point process. Our approach is based on moment identities for stopping sets, showing that the random count of points inside the complement S̅ of a stopping set S has a Poisson distribution conditionally to S. The proofs do not require the use of set-indexed martingales, and our estimators have a lower variance when compared to standard sampling. Numerical simulations are presented for examples such as the convex hull and the Voronoi flower of a Poisson point process, and their complements. [ABSTRACT FROM AUTHOR]