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  "body": "\n## Statement\n\nLet $(X, \\Sigma)$ be a [[measurable space|N4KDT83V3S]] and let $(\\mu_n)_{n \\ge 1}$ be a sequence of [[complex measures|NY6TYHRGMS]] on $\\Sigma$.\nAssume that for every $E \\in \\Sigma$ the limit $\\mu(E) = \\lim_{n \\to \\infty} \\mu_n(E)$ exists in $\\C$.\nThen:\n\n1. The $\\mu_n$ are uniformly countably additive: for every sequence $E_1 \\supseteq E_2 \\supseteq \\cdots$ in $\\Sigma$ with $\\bigcap_k E_k = \\emptyset$, one has $\\sup_n \\abs{\\mu_n(E_k)} \\to 0$ as $k \\to \\infty$.\n2. The set function $\\mu$ is [[countably additive|D7KB9HFCGZ]], so it is a complex measure on $\\Sigma$.\n\n## Remarks\n\nOne proof reduces to the Vitali–Hahn–Saks theorem.\nPut $\\lambda = \\sum_n 2^{-n} \\abs{\\mu_n} / (1 + \\abs{\\mu_n}(X))$, a finite measure.\nEach $\\mu_n$ is absolutely continuous with respect to $\\lambda$, and the Vitali–Hahn–Saks theorem gives the conclusion.\n\nExample: let $f_n$ be integrable functions on a measure space with $\\int_E f_n$ convergent for every measurable $E$.\nThen the limit $E \\mapsto \\lim_n \\int_E f_n$ is again countably additive.\n\n## Explanations\n\n### L1 human\n\nIf a sequence of measures has a limit on each measurable set, the limit is a measure too.\nThis is surprising, because pointwise limits usually lose properties such as countable additivity.\n\n### L2 human\n\nThe theorem is often stated together with the Nikodym boundedness theorem, which says that a family of complex measures bounded on each set is uniformly bounded.\nBoth are measure-theoretic forms of the uniform boundedness principle.\nThey are basic tools in the study of weak convergence in spaces of measures and in $L^1$.\n",
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    "name": "Nikodym convergence theorem",
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  "msc": [
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  "name": "Nikodym convergence theorem",
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