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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $\\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$, and let $A \\subseteq \\R^n$ be a Lebesgue measurable set with $\\lambda(A) \u003c \\infty$.\nLet $f \\colon A \\to \\R$ be a [[measurable function|YYNCJSR0PJ]].\nThen for every $\\varepsilon \u003e 0$ there is a [[compact set|0R6EWFKRCF]] $K \\subseteq A$ with $\\lambda(A \\setminus K) \u003c \\varepsilon$ such that the restriction $f|_K$ is [[continuous|0X3WGX95FE]].\n\n## Remarks\n\nOne proof approximates $f$ by simple functions and uses that each measurable set is nearly closed.\nFor $f = \\mathbf{1}_B$, choose closed $F_1 \\subseteq B$ and $F_2 \\subseteq A \\setminus B$ with small leftover measure; $f$ is constant on each of the disjoint closed sets $F_1$ and $F_2$, so it is continuous on $F_1 \\cup F_2$.\nFor general $f$, take a sequence of simple functions that converges to $f$ pointwise, and use Egorov's theorem to make the convergence uniform on a large compact set.\nExample: $f = \\mathbf{1}_{\\Q}$ on $A = [0, 1]$ is discontinuous at every point, but its restriction to a compact set $K$ of irrational numbers with $\\lambda(K) \u003e 1 - \\varepsilon$ is the constant $0$.\nThe conclusion is about $f|_K$, not about $f$: the points of $K$ need not be points of continuity of $f$.\nBy the Tietze extension theorem, there is also a continuous $g \\colon \\R^n \\to \\R$ with $g = f$ on $K$.\nThe same result holds for every Radon measure on a locally compact Hausdorff space, for functions on sets of finite measure.\n\n## Explanations\n\n### L1 human\n\nA measurable function can be very irregular, such as a function that jumps between $0$ and $1$ on the rational and irrational numbers.\nLusin's theorem says that after removing a set of small measure, what is left of the function is continuous.\nSo every measurable function is continuous up to an arbitrarily small set.\n\n### L2 human\n\nThis is the second of Littlewood's three principles.\nIt explains why continuous functions with compact support are dense in $L^p$ spaces for $1 \\le p \u003c \\infty$, together with the density of simple functions.\nIt is a basic tool for passing from statements about continuous functions to statements about measurable ones.\nIn the other direction, a function with this property for every $\\varepsilon$ is measurable, so the property characterises measurability on sets of finite measure.\n",
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