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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]].\nLet $\\mathcal{P}(X)$ be the [[power set|00E0YZCERH]] of $X$.\nThe *inner measure* induced by $\\mu$ is the map $\\mu_* \\colon \\mathcal{P}(X) \\to [0, \\infty]$ given by\n$$\\mu_*(A) = \\sup \\set{ \\mu(B) : B \\in \\Sigma, \\ B \\subseteq A },$$\nthe [[supremum|0R3H5PANRQ]] of the measures of the measurable subsets of $A$.\n\n## Remarks\n\nThe outer measure induced by $\\mu$ is $\\mu^*(A) = \\inf \\set{ \\mu(B) : B \\in \\Sigma, \\ A \\subseteq B }$, and $\\mu_*(A) \\le \\mu^*(A)$ for all $A$.\nIf $A \\in \\Sigma$, then $\\mu_*(A) = \\mu(A) = \\mu^*(A)$.\nFor Lebesgue measure $\\lambda$ on $\\R^n$, the supremum may be taken over compact subsets of $A$.\nFor Lebesgue measure and a set $A$ with $\\lambda^*(A) \u003c \\infty$, the set $A$ is Lebesgue measurable if and only if $\\lambda_*(A) = \\lambda^*(A)$.\nExample: for $A = \\Q \\cap [0, 1]$ and Lebesgue measure, $A$ is measurable with $\\lambda_*(A) = \\lambda^*(A) = 0$.\nNon-example of measurability: a Vitali set $V \\subseteq [0, 1]$ has $\\lambda_*(V) = 0 \u003c \\lambda^*(V)$.\nUnlike an outer measure, an inner measure is superadditive: $\\mu_*(A \\cup B) \\ge \\mu_*(A) + \\mu_*(B)$ for disjoint $A$ and $B$.\nSome authors define inner measures abstractly by such axioms, without a given measure.\n",
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