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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]], and let $A_1, A_2, \\ldots$ be sets in $\\Sigma$.\n\n1. (Continuity from below.) If $A_1 \\subseteq A_2 \\subseteq \\cdots$, then $\\mu\\left(\\bigcup_{n=1}^{\\infty} A_n\\right) = \\lim_{n \\to \\infty} \\mu(A_n)$.\n2. (Continuity from above.) If $A_1 \\supseteq A_2 \\supseteq \\cdots$ and $\\mu(A_1) \u003c \\infty$, then $\\mu\\left(\\bigcap_{n=1}^{\\infty} A_n\\right) = \\lim_{n \\to \\infty} \\mu(A_n)$.\n\nThe [[limits|05M5SWSZZD]] are taken in $[0, \\infty]$.\n\n## Remarks\n\nFor part 1, put $B_1 = A_1$ and $B_n = A_n \\setminus A_{n-1}$ for $n \\ge 2$.\nThe sets $B_n$ are pairwise disjoint, their union is $\\bigcup_n A_n$, and $B_1 \\cup \\cdots \\cup B_n = A_n$.\n[[Countable additivity|D7KB9HFCGZ]] gives $\\mu(\\bigcup_n A_n) = \\sum_n \\mu(B_n)$, and the partial sums of this series are the numbers $\\mu(A_n)$.\nFor part 2, apply part 1 to the increasing sets $A_1 \\setminus A_n$, and subtract from the finite number $\\mu(A_1)$.\n\nFor Lebesgue measure $\\lambda$ on $\\R$ and a point $x$, the sets $[x - 1/n, x + 1/n]$ decrease to $\\set{x}$ and have measure $2/n$.\nPart 2 gives $\\lambda(\\set{x}) = 0$.\n\nThe hypothesis $\\mu(A_1) \u003c \\infty$ in part 2 is needed.\nFor Lebesgue measure, the sets $A_n = [n, \\infty)$ decrease to $\\varnothing$, but $\\lambda(A_n) = \\infty$ for every $n$.\n\n## Explanations\n\n### L1 human\n\nIf sets grow and fill out a limit set, their measures grow to the measure of the limit set.\nIf sets shrink to a limit set, their measures shrink to its measure, provided one starts from a set of finite measure.\nSo measures behave well under monotone limits.\n\n### L2 human\n\nFor a finitely additive measure on a σ-algebra, continuity from below is equivalent to countable additivity.\nSo it is a useful way to check that a set function is a measure.\nIt is the set version of the monotone convergence theorem.\nFor [[probability measures|0X8HBKCSYB]] both parts hold without extra hypothesis, since every set has finite measure.\nContinuity of measures underlies the Borel–Cantelli lemmas and the right-continuity of distribution functions.\n",
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