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  "body": "\n## Definition\n\nLet $(X, \\Sigma)$ be a [[measurable space|N4KDT83V3S]].\nA *signed measure* on $(X, \\Sigma)$ is a map $\\nu \\colon \\Sigma \\to [-\\infty, \\infty]$, with values in the [[extended real line|0GSZ7B0CGR]], such that:\n\n1. $\\nu(\\emptyset) = 0$;\n2. $\\nu$ takes at most one of the two values $+\\infty$ and $-\\infty$;\n3. $\\nu$ is [[countably additive|D7KB9HFCGZ]]: for pairwise disjoint $E_1, E_2, \\ldots$ in $\\Sigma$,\n$$\\nu\\Big( \\bigcup_{n} E_n \\Big) = \\sum_{n} \\nu(E_n),$$\nwhere the series converges absolutely when the left side is finite.\n\n## Remarks\n\nExample: let $\\mu$ be a measure on $(X, \\Sigma)$ and $f \\in L^1(\\mu)$ real-valued.\nThen $\\nu(E) = \\int_E f \\, d\\mu$ is a signed measure with finite values.\nCountable additivity follows from the dominated convergence theorem, and absolute convergence from $\\sum_n \\abs{\\nu(E_n)} \\le \\int \\abs{f} \\, d\\mu$.\nMore generally, $\\mu_1 - \\mu_2$ is a signed measure whenever $\\mu_1, \\mu_2$ are measures and at least one of them is finite.\n\nNon-example: on $X = \\set{a, b}$ with all subsets measurable, no signed measure has $\\nu(\\set{a}) = +\\infty$ and $\\nu(\\set{b}) = -\\infty$.\nClause 2 excludes it, and indeed $\\nu(\\set{a, b}) = \\nu(\\set{a}) + \\nu(\\set{b})$ would be $\\infty - \\infty$.\n\nA *finite* signed measure is one with values in $\\R$.\nEvery measure is a signed measure.\n",
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