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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F})$ be a [[measurable space|N4KDT83V3S]] and let $n \\ge 1$.\nA *set-valued measure* on $(\\Omega, \\mathcal{F})$ with values in $\\R^n$ is a map $M$ from $\\mathcal{F}$ to the nonempty subsets of $\\R^n$ such that:\n\n1. $M(\\emptyset) = \\set{0}$.\n2. For every sequence of pairwise disjoint sets $A_1, A_2, \\dots$ in $\\mathcal{F}$ with union $A$,\n$$M(A) = \\set{\\sum_{i=1}^{\\infty} x_i : x_i \\in M(A_i) \\text{ for all } i, \\ \\sum_{i=1}^{\\infty} \\abs{x_i} \u003c \\infty}.$$\n\n## Remarks\n\nMany authors also require the values to be closed, compact or convex, and some work in a Banach space instead of $\\R^n$.\nExample: let $\\mu$ be a finite measure on $\\mathcal{F}$, let $K \\subseteq \\R^n$ be nonempty, compact and convex, and let $M(A) = \\mu(A) K$.\nThen $M(\\emptyset) = \\set{0}$.\nGiven $x_i \\in \\mu(A_i) K$, the series converges absolutely since $K$ is bounded, and its sum lies in $\\mu(A) K$ because $K$ is convex and closed.\nConversely $\\mu(A) k = \\sum_i \\mu(A_i) k$ for $k \\in K$, so clause 2 holds.\nNon-example: let $\\Omega = \\set{a, b}$ with counting measure $\\mu$ and $K = \\set{0, 1}$, which is not convex.\nThen $M(\\Omega) = 2K = \\set{0, 2}$, while the sums from $M(\\set{a})$ and $M(\\set{b})$ form $\\set{0, 1, 2}$, so clause 2 fails.\nA [[vector measure|PS28BSAJKJ]] $\\nu$ gives the set-valued measure $A \\mapsto \\set{\\nu(A)}$.\n",
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