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  "body": "\n## Summary\n\nLet $\\nu$ be a [[signed measure|VG2P7R91Z7]] or a [[complex measure|NY6TYHRGMS]].\nIts total variation $\\abs{\\nu}$ is the smallest positive measure with $\\abs{\\nu(E)} \\le \\abs{\\nu}(E)$ for every set $E$.\nIt is computed as the supremum of $\\sum \\abs{\\nu(E_i)}$ over partitions of a set into measurable pieces.\nFor a signed measure it equals $\\nu^+ + \\nu^-$ from the Jordan decomposition.\nThe number $\\abs{\\nu}(X)$ is the total variation norm, which makes the finite signed or complex measures a Banach space.\nIt extends the [[total variation|8QA5DEWNYD]] of a function to measures.\n\n## Explanations\n\n### L1 human\n\nA signed measure can make a set look small by cancelling positive and negative parts.\nThe total variation removes this cancellation: it counts all the positive mass and all the negative mass.\nFor example, on $[-1, 1]$ the signed measure $\\nu(E) = \\int_E x \\, dx$ gives the whole interval measure $0$, but its total variation gives it $\\int_{-1}^{1} \\abs{x} \\, dx = 1$.\n",
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