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  "body": "\n## Statement\n\nLet $(X, \\Sigma)$ be a measurable space and let $\\mu_1, \\dots, \\mu_n$ be finite real [[signed measures|VG2P7R91Z7]] on $\\Sigma$.\nAssume that each $\\mu_i$ is [[non-atomic|SV96KN4WR9]], that is, its total variation $\\abs{\\mu_i}$ has no atoms.\nLet $\\mu \\colon \\Sigma \\to \\R^n$ be the [[vector measure|PS28BSAJKJ]] $\\mu(E) = (\\mu_1(E), \\dots, \\mu_n(E))$.\nThen the range $\\set{\\mu(E) : E \\in \\Sigma}$ is a [[compact|0R6EWFKRCF]] [[convex|02AS45T3H9]] subset of $\\R^n$.\n\n## Remarks\n\nA short proof considers the set $K$ of measurable $g \\colon X \\to [0, 1]$, compact in the weak-$*$ topology of $L^\\infty(\\abs{\\mu_1} + \\dots + \\abs{\\mu_n})$.\nThe map $g \\mapsto \\int g \\, d\\mu$ is linear and continuous, so its image is compact and convex, and it contains the range of $\\mu$.\nFor a point $v$ of the image, the preimage of $v$ in $K$ is compact and convex and has an extreme point by the Krein–Milman theorem.\nNon-atomicity forces such an extreme point to be an indicator function $\\mathbf{1}_E$, so $v = \\mu(E)$.\n\nExample: for $n = 1$ and Lebesgue measure on $[0, 1]$, the range is $[0, 1]$, because $\\lambda([0, t]) = t$.\n\nWithout non-atomicity the result fails: the Dirac measure $\\delta_0$ has range $\\{0, 1\\}$, which is not convex.\nThe result also fails in infinite dimensions.\nThe vector measure $E \\mapsto \\mathbf{1}_E$ from the Borel sets of $[0, 1]$ to $L^2[0, 1]$ is non-atomic, but $\\frac12 \\mathbf{1}_{[0,1]}$ lies between $0$ and $\\mathbf{1}_{[0,1]}$ and is not an indicator function.\n\n## Explanations\n\n### L1 human\n\nSuppose several quantities are spread over a space without any point carrying a lump.\nThen every proportion of these quantities can be cut out exactly by one piece of the space.\nFor instance, a cake with chocolate and cream spread smoothly can be cut so that one piece has exactly a third of the chocolate and exactly half of the cream.\n\n### L2 human\n\nThe theorem is the basis of fair division results, such as the existence of exact divisions of a cake among several players.\nIn control theory it gives the bang-bang principle: the set of states reachable with measurable controls is already reached with controls taking only extreme values.\nIn economics it shows that non-atomic economies have convex aggregate behaviour.\n",
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