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  "body": "\n## Definition\n\nLet $X$ be a [[set|0SJ3RX9TXA]] and let $\\mathcal{F}$ be a family of subsets of $X$.\nLet $\\mu \\colon \\mathcal{F} \\to [0, \\infty]$ be a function with values in the [[extended real number line|0GSZ7B0CGR]].\nThe function $\\mu$ is *countably additive*, or *σ-additive*, if the following holds.\nFor every sequence $A_1, A_2, \\ldots$ of pairwise disjoint sets in $\\mathcal{F}$ whose [[union|07KVB4JSC7]] is in $\\mathcal{F}$,\n$$\\mu\\left(\\bigcup_{n=1}^{\\infty} A_n\\right) = \\sum_{n=1}^{\\infty} \\mu(A_n).$$\n\n## Remarks\n\nOn the power set of $\\N$, the function $\\mu(A) = \\sum_{n \\in A} 2^{-n}$ is countably additive.\nThe [[series|0NM626FH6Q]] has non-negative terms, so its sum does not depend on how the terms are grouped.\nOn the family $\\mathcal{F}$ of subsets $A \\subseteq \\N$ such that $A$ or $\\N \\setminus A$ is finite, let $\\nu(A) = 0$ for finite $A$ and $\\nu(A) = 1$ otherwise.\nThis $\\nu$ is finitely additive but not countably additive.\nIndeed $\\N = \\bigcup_{n} \\set{n}$ is a disjoint union of sets in $\\mathcal{F}$, but $\\nu(\\N) = 1$ and $\\sum_n \\nu(\\set{n}) = 0$.\nIf $\\varnothing \\in \\mathcal{F}$ and $\\mu$ takes at least one finite value, then countable additivity forces $\\mu(\\varnothing) = 0$: take $A_1 = A$ with $\\mu(A) \u003c \\infty$ and $A_n = \\varnothing$ for $n \\ge 2$.\nThe same definition is used for functions with values in $\\R$, $\\C$ or a Banach space; then the series must converge.\n",
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