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  "body": "\n## Summary\n\nTwo [[measures|PR11BZQQPG]] on the same space are mutually singular when they live on disjoint parts of the space.\nOne of them gives zero mass to a set, and the other gives zero mass to its complement.\nMutual singularity is the opposite extreme of [[absolute continuity of measures|P0P21KDV6E]].\nThe [[Lebesgue decomposition theorem|74VMD07DRX]] splits a σ-finite measure, relative to a σ-finite reference measure, into an absolutely continuous part and a singular part.\n\n## Explanations\n\n### L1 human\n\nPicture two piles of sand on the same floor that never touch.\nEach pile sits on its own region, and the two regions do not overlap.\nOn the real line, Lebesgue measure spreads mass evenly, while the point mass at $0$ puts all its mass on the single point $0$.\nLebesgue measure gives the set $\\{0\\}$ length zero, and the point mass gives the rest of the line mass zero.\nSo these two measures are mutually singular.\n",
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