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  "body": "\n## Summary\n\nA non-measurable set is a set to which a given measure cannot assign a size, because it lies outside the $\\sigma$-algebra of measurable sets.\nThe term most often refers to subsets of $\\R^n$ that are not measurable for [[Lebesgue measure|2Z2QVWZ262]].\nTheir existence shows that length, area and volume cannot be defined for all sets in a way that is countably additive and invariant under translations.\nTheir construction needs some form of the [[axiom of choice|0HTGB13FE2]].\nSolovay showed that ZF with only dependent choice cannot prove that they exist, if an inaccessible cardinal is consistent with ZFC.\n\n## Explanations\n\n### L1 human\n\nLength works well for intervals and for all sets one usually meets.\nBut with the axiom of choice one can build sets of real numbers so scattered that no length fits them.\nThe classical example is a [[Vitali set|WYHGXY471C]], which picks one number from each class of numbers that differ by a fraction.\nShifted copies of it would have to have equal lengths that add up to a finite positive number, which no single length can do.\n",
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