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AbstractThe interpretation of the ‘standard’ Palm version of a stationary random measure ? is that it behaves like ? conditioned on containing the origin in its mass. The interpretation of the ‘modified’ Palm version is that it behaves like ? seen from a typical location in its mass. In this paper we shall focus on the modified Palm version, comparing it with the standard version in the transparent case of mixed biased coin tosses, and then establishing a limit theorem that motivates the above interpretation in the case of random measures on locally compact second countable Abelian groups possessing Følner averaging sets.