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  "body": "\n## Definition\n\nLet $\\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\\R$, and let \"nearly\" mean \"outside a set of arbitrarily small measure\".\n*Littlewood's three principles* are the following three heuristic statements:\n\n1. Every Lebesgue measurable set of finite measure is nearly a finite union of intervals.\n2. Every [[measurable function|YYNCJSR0PJ]] is nearly [[continuous|0X3WGX95FE]].\n3. Every pointwise convergent sequence of measurable functions is nearly [[uniformly convergent|08VZ57PT26]].\n\n## Remarks\n\nThe precise forms on a set $E \\subseteq \\R$ with $\\lambda(E) \u003c \\infty$ are as follows.\nFor principle 1: for every $\\varepsilon \u003e 0$ there is a finite union $U$ of intervals with $\\lambda(E \\mathbin{\\triangle} U) \u003c \\varepsilon$.\nFor principle 2: this is [[Lusin's theorem|GZMAH9W2SK]], which gives a closed $F \\subseteq E$ with $\\lambda(E \\setminus F) \u003c \\varepsilon$ on which $f$ is continuous.\nFor principle 3: this is [[Egorov's theorem|762Y2NYA7W]], which gives uniform convergence outside a set of measure less than $\\varepsilon$.\nExample: for $f = \\mathbf{1}_{\\Q}$ on $[0, 1]$, take $F$ a closed set of irrationals with $\\lambda(F) \u003e 1 - \\varepsilon$; then $f|_F = 0$ is continuous.\nWithout finite measure, principle 3 fails: on $\\R$, the functions $\\mathbf{1}_{[n, \\infty)}$ tend to $0$ pointwise, but not uniformly outside any set of finite measure.\n\n## Explanations\n\n### L1 human\n\nThe three principles say that measure theory is close to ordinary analysis, as long as one is allowed to ignore a small set.\nSets are almost intervals, functions are almost continuous, and convergence is almost uniform.\n",
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