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We discuss the following question: Let $V$ be the void functional of a random closed set. For which $\alpha>0$ is $V^\alpha$ a void functional? We answer this question when $V$ is the void functional of a random subset of a finite set. The result is then generalized to exponents which preserve complete monotonicity of functions on finite lattices. The results are analogous to the result of FitzGerald and Horn on Hadamard powers of positive semi-definite matrices. Also, we study the question of approximating an $m$-divisible random set by infinitely divisible random sets. The result is analogous to that of Arak's classical result on approximating an $m$-divisible random variable by infinitely divisible random variables. |