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  "body": "\n## Summary\n\nA quasi-measure-preserving map is a measurable map between measure spaces under which the preimage of every null set is a null set.\nEquivalently, the pushforward of the first measure is [[absolutely continuous|P0P21KDV6E]] with respect to the second one.\nSuch maps respect \"almost everywhere\" statements, so they act on equivalence classes of measurable functions.\nEvery [[measure-preserving transformation|556WBDAHNP]] is quasi-measure-preserving.\n\n## Explanations\n\n### L1 human\n\nA measure-preserving map keeps the size of every set.\nA quasi-measure-preserving map is allowed to change sizes, but it may not turn something of size zero into something of positive size when one pulls back.\nFor example, the map $x \\mapsto 2x$ on the real line doubles lengths, so it does not preserve length.\nStill, the preimage of a set of length zero has length zero, so the map is quasi-measure-preserving.\n",
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