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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a measure space.\nLet $f_1, f_2, \\dots \\colon X \\to [0, \\infty]$ be measurable functions.\nThen $\\liminf_{n} f_n$ is measurable, and\n$$\\int_X \\liminf_{n \\to \\infty} f_n \\, d\\mu \\le \\liminf_{n \\to \\infty} \\int_X f_n \\, d\\mu,$$\nwhere $\\liminf$ is the [[limit inferior|0H4R3RESGV]] and the integrals are [[integrals of non-negative functions|DG76G5Z6XW]].\n\n## Remarks\n\nProof idea: let $g_k = \\inf_{n \\ge k} f_n$.\nThe functions $g_k$ increase to $\\liminf_n f_n$, and $\\int g_k \\le \\int f_n$ for every $n \\ge k$.\nThe [[monotone convergence theorem|G2S6H789RG]] gives $\\int \\liminf_n f_n = \\lim_k \\int g_k \\le \\liminf_n \\int f_n$.\n\nThe inequality can be strict.\nFor $f_n = n \\mathbf{1}_{(0, 1/n)}$ on $\\R$ with Lebesgue measure, $f_n \\to 0$ pointwise but $\\int f_n = 1$ for all $n$.\n\nNon-negativity is needed.\nFor $f_n = -\\mathbf{1}_{[n, n+1]}$ on $\\R$, the left side is $0$ and the right side is $-1$.\n\n## Explanations\n\n### L1 human\n\nWhen a sequence of non-negative functions converges, mass can escape, but it cannot appear from nowhere.\nSo the integral of the limit is at most the limit of the integrals.\nA tall thin spike that keeps its area while it shrinks is the typical way mass escapes.\n\n### L2 human\n\nFatou's lemma is the step between the monotone convergence theorem and the dominated convergence theorem.\nIt is often used to show that a limit function is integrable: if $\\int f_n \\le C$ for all $n$, then $\\int \\liminf f_n \\le C$.\nIt also shows that the $L^p$ norm is lower semicontinuous under pointwise convergence almost everywhere.\n",
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