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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]] such that $\\mu$ is [[non-atomic|SV96KN4WR9]].\nLet $A \\in \\Sigma$ with $\\mu(A) \u003c \\infty$, and let $b$ be a real number with $0 \\le b \\le \\mu(A)$.\nThen there is a set $B \\in \\Sigma$ with $B \\subseteq A$ and $\\mu(B) = b$.\n\n## Remarks\n\nThe first step shows that a set of positive finite measure contains measurable subsets of arbitrarily small positive measure.\nThis uses the absence of atoms to split a set again and again into pieces of positive measure, keeping the smaller piece.\nThe second step builds $B$ by adding, at each stage, a subset of nearly the largest size that keeps the total at most $b$.\nContinuity of measure then shows that the limit set has measure exactly $b$.\n\nFor Lebesgue measure on $[0, 1]$ the conclusion is easy: take $B = [0, b]$.\nThe theorem matters for abstract non-atomic measures, such as the product measure of fair coin tosses on $\\set{0, 1}^{\\N}$.\n\nThe hypothesis that $\\mu$ is non-atomic is needed.\nFor the [[Dirac measure|0ACRGYNHSR]] $\\delta_x$ and $A = \\set{x}$, every measurable subset of $A$ has measure $0$ or $1$, so no subset has measure $1/2$.\n\n## Explanations\n\n### L1 human\n\nA measure without lumps can be cut into a piece of any size one likes.\nIf a set has measure $5$, one can find a part of it with measure exactly $\\pi$, or $0.3$, or any number between $0$ and $5$.\nLumpy measures, such as point masses, cannot be cut like this.\n\n### L2 human\n\nAs a corollary, a finite non-atomic measure takes every value in $[0, \\mu(X)]$, so its range is an interval.\nThe Lyapunov convexity theorem extends this to finite non-atomic vector measures with values in $\\R^n$: their range is compact and convex.\nThe theorem is used in probability to build events of any prescribed probability, and in economics and game theory for fair division of non-atomic resources.\n",
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    "name": "Sierpiński's theorem on non-atomic measures",
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  "markdown": "https://math2.tamnd.com/node/J76687HP2E.md",
  "msc": [
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  "name": "Sierpiński's theorem on non-atomic measures",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/J76/J76687HP2E.md",
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  "url": "https://math2.tamnd.com/wiki/sierpinskis-theorem-on-non-atomic-measures/",
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