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  "body": "\n## Summary\n\nThe pushforward measure moves a [[measure|PR11BZQQPG]] along a [[measurable function|YYNCJSR0PJ]] from one space to another.\nThe new measure gives a set the old measure of its preimage.\nIn probability theory, the distribution of a random variable is the pushforward of the probability measure along the variable.\nIt also gives the abstract change of variables formula for integrals.\n\n## Explanations\n\n### L1 human\n\nThink of the measure as sand spread over a space $X$, and of the map $f$ as a rule that moves each grain to a point of $Y$.\nThe pushforward is the new pile of sand on $Y$.\nTo weigh a region $B$ of $Y$, weigh all the grains of $X$ that land in $B$.\nFor example, let $X = \\set{1, 2, 3, 4, 5, 6}$ with mass $1/6$ at each point, and let $f(x) = 0$ for even $x$ and $f(x) = 1$ for odd $x$.\nThe pushforward puts mass $1/2$ at $0$ and mass $1/2$ at $1$.\n",
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