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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $f \\colon [a, b] \\to \\R$ be a function.\nFor $U \\colon [a, b] \\to \\R$ and $x \\in [a, b]$, the lower and upper derivatives of $U$ at $x$ are\n$$\\underline{D} U(x) = \\liminf_{y \\to x} \\frac{U(y) - U(x)}{y - x}, \\qquad \\overline{D} U(x) = \\limsup_{y \\to x} \\frac{U(y) - U(x)}{y - x},$$\nwhere $y$ ranges over $[a, b] \\setminus \\set{x}$.\nA *major function* of $f$ is a function $U$ with $\\underline{D} U(x) \\ne -\\infty$ and $\\underline{D} U(x) \\ge f(x)$ for every $x \\in [a, b]$.\nA *minor function* of $f$ is a function $V$ with $\\overline{D} V(x) \\ne +\\infty$ and $\\overline{D} V(x) \\le f(x)$ for every $x \\in [a, b]$.\nLet $\\mathcal{U}$ be the set of major functions and $\\mathcal{V}$ the set of minor functions of $f$.\nThe function $f$ is *Perron integrable* on $[a, b]$ if both sets are nonempty and\n$$\\inf_{U \\in \\mathcal{U}} \\bigl(U(b) - U(a)\\bigr) = \\sup_{V \\in \\mathcal{V}} \\bigl(V(b) - V(a)\\bigr) \\in \\R.$$\nThe *Perron integral* of $f$ is this value.\n\n## Remarks\n\nFor any major function $U$ and minor function $V$, the difference $U - V$ is nondecreasing, so $V(b) - V(a) \\le U(b) - U(a)$ always holds.\n\nExample: if $F$ is differentiable at every point of $[a, b]$ and $f = F'$, then $F$ is both a major and a minor function of $f$.\nSo $f$ is Perron integrable with integral $F(b) - F(a)$, even when $f$ is not Lebesgue integrable.\n\nNon-example: $f(x) = 1/x$ for $0 \u003c x \\le 1$ and $f(0) = 0$ is not Perron integrable on $[0, 1]$.\nA nonnegative Perron integrable function is Lebesgue integrable, and $\\int_0^1 dx / x = \\infty$.\n\nA function is Perron integrable if and only if it is [[Henstock–Kurzweil integrable|FCGHXDGVV9]], and the two integrals agree.\n",
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