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  "body": "\n## Definition\n\nLet $(X, \\Sigma)$ be a [[measurable space|N4KDT83V3S]].\nA *complex measure* on $(X, \\Sigma)$ is a map $\\nu \\colon \\Sigma \\to \\C$ with the following property.\nFor every sequence $(E_n)$ of pairwise disjoint sets in $\\Sigma$,\n$$\\nu\\Big( \\bigcup_{n} E_n \\Big) = \\sum_{n} \\nu(E_n),$$\nwhere the series of [[complex numbers|09FTAET5Z4]] converges.\n\n## Remarks\n\nTaking all $E_n = \\emptyset$ gives $\\nu(\\emptyset) = 0$.\nThe left side does not depend on the order of the $E_n$, so every rearrangement of the series converges to the same value, and the series converges absolutely.\n\nExample: let $\\mu$ be a measure and $f \\in L^1(\\mu)$ complex-valued.\nThen $\\nu(E) = \\int_E f \\, d\\mu$ is a complex measure; countable additivity follows from the dominated convergence theorem with dominating function $\\abs{f}$.\n\nNon-example: Lebesgue measure on $\\R$ is not a complex measure, because $\\lambda(\\R) = \\infty$ is not a complex number.\n\nWriting $\\nu = \\nu_r + i \\nu_i$, the real and imaginary parts $\\nu_r$ and $\\nu_i$ are finite [[signed measures|VG2P7R91Z7]].\nIntegration against $\\nu$ is defined by $\\int f \\, d\\nu = \\int f \\, d\\nu_r + i \\int f \\, d\\nu_i$.\n",
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