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  "body": "\n## Summary\n\nThe inner measure of a set approximates it from inside: it is the largest measure of a measurable set contained in it.\nIt is the companion of the [[outer measure|BJ314H17HC]], which approximates from outside.\nFor Lebesgue measure, a set of finite outer measure is measurable exactly when its inner and outer measures agree.\nHistorically, Lebesgue defined measurability in this way.\n\n## Explanations\n\n### L1 human\n\nTo measure an odd-shaped region from inside, fill it with pieces whose size is known, and take the largest total size you can fit.\nThe outer measure does the opposite: it covers the region and takes the smallest cover.\nFor a nice region the two numbers agree.\nFor example, the set of irrational numbers in $[0, 1]$ has inner and outer Lebesgue measure $1$.\nFor a Vitali set in $[0, 1]$, the inner measure is $0$ and the outer measure is positive.\n",
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