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  "body": "\n## Summary\n\nA measure on a topological space is inner regular when each measurable set is approximated from inside by compact subsets.\nThe approximation is in measure.\nInner regularity lets one reduce questions about general sets to questions about compact sets, where topology gives control.\nIt is one of the two halves of the definition of a Radon measure.\nSome authors use closed sets instead of compact sets in the definition.\n\n## Explanations\n\n### L1 human\n\nTo measure a complicated set, fill it from inside with compact pieces, like filling a lake with larger and larger islands.\nIf the islands can capture as much of the measure as one likes, the measure is inner regular.\nFor length on the real line, the set of irrational numbers in $[0, 1]$ has length $1$.\nIt contains compact sets, built by removing small open intervals around the rationals, of length as close to $1$ as one likes.\n",
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