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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[algebra of sets|PFMGZ7988V]] on a set $X$.\nA *finitely additive measure* on $\\mathcal{A}$ is a function $\\mu \\colon \\mathcal{A} \\to [0, \\infty]$, with values in the [[extended real number line|0GSZ7B0CGR]], such that:\n\n1. $\\mu(\\varnothing) = 0$.\n2. $\\mu(A \\cup B) = \\mu(A) + \\mu(B)$ for all $A, B \\in \\mathcal{A}$ with $A \\cap B = \\varnothing$.\n\n## Remarks\n\nLet $\\mathcal{A}$ be the algebra of subsets $A \\subseteq \\N$ such that $A$ or $\\N \\setminus A$ is finite.\nLet $\\nu(A) = 0$ for finite $A$ and $\\nu(A) = 1$ otherwise.\nClause 1 holds, and clause 2 holds since two disjoint sets cannot both have finite complement.\nThis $\\nu$ is not [[countably additive|D7KB9HFCGZ]], since $\\nu(\\N) = 1$ while $\\nu(\\set{n}) = 0$ for each $n$.\nOn the same algebra, the function with $\\mu(\\varnothing) = 0$ and $\\mu(A) = 1$ for $A \\ne \\varnothing$ is not finitely additive, since clause 2 fails for $A = \\set{1}$ and $B = \\set{2}$.\nBy induction, clause 2 gives additivity over any finite disjoint union.\nA finitely additive measure is monotone: $\\mu(A) \\le \\mu(B)$ when $A \\subseteq B$.\nOther names are *content* and *charge*; the word content is also used for the same notion on a ring or a semiring of sets.\n",
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