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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a measure space, and let $f_1, f_2, \\ldots$ and $f$ be measurable functions from $X$ to $\\R$.\nIf $f_n \\to f$ [[in measure|G5WT6QN8WR]], then there is a [[subsequence|0F1PR74BRX]] $(f_{n_k})$ such that $f_{n_k}(x) \\to f(x)$ for [[almost every|RARTMA0ZEC]] $x \\in X$.\n\n## Remarks\n\nChoose $n_1 \u003c n_2 \u003c \\cdots$ with $\\mu(E_k) \u003c 2^{-k}$, where $E_k = \\set{x : \\abs{f_{n_k}(x) - f(x)} \\ge 2^{-k}}$.\nLet $E = \\bigcap_{j} \\bigcup_{k \\ge j} E_k$; then $\\mu(E) \\le \\sum_{k \\ge j} 2^{-k} = 2^{1-j}$ for every $j$, so $\\mu(E) = 0$.\nFor $x \\notin E$, one has $\\abs{f_{n_k}(x) - f(x)} \u003c 2^{-k}$ for all large $k$, so $f_{n_k}(x) \\to f(x)$.\nExample: the sliding indicator functions $f_{2^k + j} = \\mathbf{1}_{[j 2^{-k}, (j+1) 2^{-k}]}$ on $[0, 1]$ converge to $0$ in measure but at no point; the subsequence $f_{2^k}$, the indicators of $[0, 2^{-k}]$, converges to $0$ at every $x \\ne 0$.\nThe passage to a subsequence is necessary, as this example shows.\n\n## Explanations\n\n### L1 human\n\nConvergence in measure is weak: it allows the sequence to fail to converge at every point.\nThis theorem says that one can always pick out a subsequence that does converge, except on a set of measure zero.\n\n### L2 human\n\nThe result is a standard tool to pass from integral estimates to pointwise statements.\nFor example, if $f_n \\to f$ in $L^p$, then $f_n \\to f$ in measure, so a subsequence converges to $f$ almost everywhere; this is a step in the proof that $L^p$ is complete.\nOn a space of finite measure, it gives a characterisation of convergence in measure.\nThere, $f_n \\to f$ in measure if and only if every subsequence has a further subsequence converging to $f$ almost everywhere.\n",
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