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  "body": "\n## Summary\n\nA Haar measure on a [[locally compact group|AHV05ZBCYW]] is a nonzero [[Radon measure|HZNEQQBCW4]] that does not change under left translation by elements of the group.\nEvery locally compact group has one, and it is unique up to a positive constant.\nIt extends Lebesgue measure on $\\R^n$ and counting measure on discrete groups, and it is the basic tool for integration and harmonic analysis on groups.\n\n## Explanations\n\n### L1 human\n\nOn the real line, the length of an interval does not change when the interval is shifted.\nA Haar measure is a way of assigning sizes to subsets of a group with the same property: moving a set by a group element keeps its size.\nOn the circle it is arc length, on a finite group it counts points, and on the positive reals under multiplication it is $dx/x$.\nIndeed, the interval $[a, b]$ has measure $\\log(b/a)$ for $dx/x$, and multiplying it by $c \u003e 0$ gives $[ca, cb]$ with the same measure.\n",
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