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  "body": "\n## Summary\n\nOn a [[Lie group|7JBRGCVJRB]], the [[Haar measure|VY0HXD3GC4]] can be written down explicitly as the measure defined by a nonzero left-invariant differential form of top degree.\nThis gives a smooth density in every chart, so integrals over the group can be computed with ordinary calculus.\nThe left and right Haar measures agree exactly when $\\abs{\\det \\operatorname{Ad}(g)} = 1$ for all $g$, which holds for compact, abelian and semisimple groups.\nIt is the basic tool for averaging over a group, as in the proof that representations of compact groups are unitarisable.\n\n## Explanations\n\n### L1 human\n\nA Haar measure is a way to measure the size of sets of group elements that does not change when the set is moved by the group.\nOn a Lie group one gets it by fixing a small volume element at the identity and carrying it to every other point by left multiplication.\nOn the circle group this gives the arc length $d\\theta$.\nOn the positive real numbers under multiplication it gives $dx/x$, which is unchanged by the substitution $x \\mapsto cx$.\n",
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