{
  "aliases": [
    "continuity-from-below",
    "continuity-from-above"
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  "body": "\n## Statement\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]].\nIf $A_1 \\subseteq A_2 \\subseteq \\cdots$ are events and $A = \\bigcup_{n=1}^{\\infty} A_n$, then $\\lim_{n \\to \\infty} P(A_n) = P(A)$.\nIf $B_1 \\supseteq B_2 \\supseteq \\cdots$ are events and $B = \\bigcap_{n=1}^{\\infty} B_n$, then $\\lim_{n \\to \\infty} P(B_n) = P(B)$.\n\n## Remarks\n\nThe first part is continuity from below. Its proof writes $A$ as the disjoint union of $A_1$ and the sets $A_{n+1} \\setminus A_n$, and uses countable additivity.\nThe second part, continuity from above, follows from the first part with the complements $B_n^c$.\nFor a general measure $\\mu$, continuity from above needs $\\mu(B_1) \u003c \\infty$. A [[probability measure|0VGMEBVW1M]] is finite, so this holds.\nFor example, $P(X \\le x + 1/n) \\to P(X \\le x)$ for a real random variable $X$. This gives the right continuity of the cumulative distribution function.\n\n## Explanations\n\n### L1 human\n\nIf events grow step by step, their probabilities grow toward the probability of the limit event.\nFor example, the chance that a fair coin shows heads at least once in the first $n$ tosses is $1 - 2^{-n}$. It tends to the chance $1$ of heads at some toss.\n",
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  "name": "Continuity of probability measures",
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