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  "body": "\n## Summary\n\nThe modular function of a [[locally compact group|AHV05ZBCYW]] measures how far a left [[Haar measure|VY0HXD3GC4]] is from being right-invariant.\nIt is a continuous homomorphism $\\Delta$ from the group to the multiplicative group of positive reals.\nIt appears in every formula that moves a Haar integral across a right translation or an inversion.\nIt is identically $1$ exactly for unimodular groups.\n\n## Explanations\n\n### L1 human\n\nA left Haar measure does not change when a set is moved on the left, but it may grow or shrink when the set is moved on the right.\nThe growth factor depends only on the element used to move, and that factor is the modular function.\nFor the group of maps $x \\mapsto ax + b$ with $a \u003e 0$, moving a set on the right by such a map multiplies its left Haar measure by $1/a$.\nFor commutative groups left and right are the same, so the factor is always $1$.\n",
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