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  "body": "\n## Statement\n\nLet $(X, \\Sigma)$ be a measurable space and let $\\nu$ be a [[signed measure|VG2P7R91Z7]] on it.\nThen there are unique [[measures|PR11BZQQPG]] $\\nu^+$ and $\\nu^-$ on $(X, \\Sigma)$ such that $\\nu^+$ and $\\nu^-$ are [[mutually singular|EBGECZNCV5]] and\n$$\\nu = \\nu^+ - \\nu^-.$$\nAt least one of $\\nu^+$ and $\\nu^-$ is finite.\n\n## Remarks\n\nExistence comes from a [[Hahn decomposition|2PKHANZ8JS]] $X = P \\cup N$: put $\\nu^+(E) = \\nu(E \\cap P)$ and $\\nu^-(E) = -\\nu(E \\cap N)$.\nThese are measures, they are concentrated on the disjoint sets $P$ and $N$, and their difference is $\\nu$.\nSince $\\nu$ takes at most one infinite value, one of them is finite.\nUniqueness: any such pair gives a Hahn decomposition, and two Hahn decompositions differ by a $\\nu$-null set.\n\nExample: for $\\nu(E) = \\int_E f \\, d\\mu$ with $f \\in L^1(\\mu)$ real, $\\nu^+(E) = \\int_E f^+ \\, d\\mu$ and $\\nu^-(E) = \\int_E f^- \\, d\\mu$, where $f^+ = \\max(f, 0)$ and $f^- = \\max(-f, 0)$.\n\nMutual singularity is what makes the decomposition unique.\nWithout it, $\\nu = (\\nu^+ + \\mu) - (\\nu^- + \\mu)$ for any finite measure $\\mu$.\n\n## Explanations\n\n### L1 human\n\nEvery signed measure is the difference between a positive part and a negative part.\nThe two parts live on separate regions of the space, so they do not cancel each other anywhere.\nIt is the measure version of writing a number as its positive part minus its negative part.\n\n### L2 human\n\nThe Jordan decomposition reduces signed measures to ordinary measures: one integrates against $\\nu$ by integrating against $\\nu^+$ and $\\nu^-$.\nThe sum $\\nu^+ + \\nu^-$ is the [[total variation|EPBND51CR5]] $\\abs{\\nu}$, the natural size of $\\nu$.\nThe name comes from the [[Jordan decomposition of a function of bounded variation|KDT3E0VE0K]] as a difference of increasing functions.\nFor the measure $dF$ of such a function $F$ the two decompositions correspond.\n",
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  "msc": [
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