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A random vector $\\xi$ in $\\mathbb{R}^n$ is called swap-invariant if $\\,{\\mathbf E}\\,\\big| \\!\\sum_j u_j \\xi_j \\big|\\,$ is invariant under all permutations of $(\\xi_1, \\ldots, \\xi_n)$ for each $u \\in \\mathbb{R}^n$. We extend this notion to random measures. For a swap-invariant random measure $\\xi$ on a measure space $(S,\\mathcal{S},\\mu)$ the vector $(\\xi(A_1), \\ldots, \\xi(A_n))$ is swap-invariant for all disjoint $A_j \\in \\mathcal{S}$ with equal $\\mu$-measure. 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