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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $\\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$, and let $f \\colon \\R^n \\to \\R$ be locally [[integrable|X7GTF3D98F]], that is, integrable on every bounded measurable set.\nFor $x \\in \\R^n$ and $r \u003e 0$, let $B(x, r)$ be the [[open ball|0YE5868HZB]] of centre $x$ and radius $r$.\nThen for [[almost every|RARTMA0ZEC]] $x \\in \\R^n$,\n$$\\lim_{r \\to 0^+} \\frac{1}{\\lambda(B(x, r))} \\int_{B(x, r)} \\abs{f(y) - f(x)} \\, d\\lambda(y) = 0.$$\nIn particular, $\\lim_{r \\to 0^+} \\frac{1}{\\lambda(B(x, r))} \\int_{B(x, r)} f \\, d\\lambda = f(x)$ for almost every $x$.\n\n## Remarks\n\nThe conclusion says that almost every point is a [[Lebesgue point|19WRHM5NAT]] of $f$.\nThe proof first notes that the claim is clear for continuous $f$ with compact support, which are dense in $L^1$.\nThe error for general $f$ is controlled by the [[Hardy–Littlewood maximal function|VX1A56P5KQ]], which satisfies the weak type bound $\\lambda(\\set{Mf \u003e t}) \\le 3^n \\norm{f}_1 / t$.\nThis bound comes from the [[Vitali covering lemma|FCA0J1X07K]].\n\nExample: for $n = 1$ and $f$ integrable on $\\R$, the theorem says $F(x) = \\int_0^x f$ satisfies $F'(x) = f(x)$ for almost every $x$.\n\nThe balls can be replaced by other families that shrink nicely, such as cubes containing $x$.\nThey cannot be replaced by arbitrary rectangles containing $x$: in dimension $n \\ge 2$, for rectangles of unbounded eccentricity the conclusion can fail for some integrable $f$.\n\n## Explanations\n\n### L1 human\n\nAverage an integrable function over smaller and smaller balls around a point.\nFor almost every point, these averages tend to the value of the function at that point.\nSo an integrable function, however rough, is determined by its local averages.\n\n### L2 human\n\nThe theorem is the measure-theoretic form of the fundamental theorem of calculus.\nIt gives Lebesgue's density theorem when $f$ is an indicator function.\nIt extends to Radon measures on $\\R^n$ through the Besicovitch covering theorem, and to doubling metric measure spaces.\n",
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