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We prove that intersections and unions of independent random sets in finitespaces achieve a form of Lipschitz continuity. More precisely, given thedistribution of a random set $\Xi$, the function mapping any random setdistribution to the distribution of its intersection (under independenceassumption) with $\Xi$ is Lipschitz continuous with unit Lipschitz constant ifthe space of random set distributions is endowed with a metric defined as the$L_k$ norm distance between inclusion functionals also known as commonalities.Moreover, the function mapping any random set distribution to the distributionof its union (under independence assumption) with $\Xi$ is Lipschitz continuouswith unit Lipschitz constant if the space of random set distributions isendowed with a metric defined as the $L_k$ norm distance between hittingfunctionals also known as plausibilities.  Using the epistemic random set interpretation of belief functions, we alsodiscuss the ability of these distances to yield conflict measures. All theproofs in this paper are derived in the framework of Dempster-Shafer belieffunctions. Let alone the discussion on conflict measures, it is straightforwardto transcribe the proofs into the general (non necessarily epistemic) randomset terminology.

Complementary Lipschitz continuity results for the distribution of intersections or unions of independent random sets in finite discrete spaces