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  "body": "\n## Definition\n\nLet $(X, \\Sigma)$ be a [[measurable space|N4KDT83V3S]].\nA subset $A \\subseteq X$ is *non-measurable* (with respect to $\\Sigma$) if $A \\notin \\Sigma$.\nWithout further qualification, a non-measurable subset of $\\R^n$ is a set that is not Lebesgue measurable.\n\n## Remarks\n\nExample: on $X = \\set{1, 2}$ with $\\Sigma = \\set{\\varnothing, X}$, the set $\\set{1}$ is non-measurable.\nExample: a Vitali set in $[0, 1]$ is not Lebesgue measurable.\nNon-example: every interval, every open set and every closed subset of $\\R^n$ is Lebesgue measurable, so none of them is non-measurable.\nFor Lebesgue measure, $A \\subseteq \\R^n$ is non-measurable if and only if there is a set $B$ with $\\lambda^*(B) \u003c \\lambda^*(B \\cap A) + \\lambda^*(B \\setminus A)$, where $\\lambda^*$ is Lebesgue outer measure.\nEvery subset of $\\R^n$ with positive outer measure contains a non-measurable subset.\nBeing non-measurable depends on $\\Sigma$: a set that is not a Borel set may still be Lebesgue measurable.\n",
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