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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a measure space and let $f \\colon X \\to \\R$ be an [[integrable function|X7GTF3D98F]].\nLet $f^+ = \\max(f, 0)$ and $f^- = \\max(-f, 0)$, so that $f = f^+ - f^-$.\nThe *Lebesgue integral* of $f$ is\n$$\\int_X f \\, d\\mu = \\int_X f^+ \\, d\\mu - \\int_X f^- \\, d\\mu,$$\nwhere both terms are [[integrals of non-negative functions|DG76G5Z6XW]] and are finite.\nFor an integrable $f \\colon X \\to \\C$, the integral is $\\int_X f \\, d\\mu = \\int_X \\operatorname{Re} f \\, d\\mu + i \\int_X \\operatorname{Im} f \\, d\\mu$.\n\n## Remarks\n\nExample: on $[-1, 1]$ with Lebesgue measure, let $f(x) = x$.\nThen $f^+ = x \\mathbf{1}_{[0, 1]}$ and $f^- = -x \\mathbf{1}_{[-1, 0]}$ both have integral $1/2$, so $\\int f = 0$.\n\nNon-example: $g(x) = x$ on all of $\\R$ is not integrable.\nBoth $\\int g^+$ and $\\int g^-$ are infinite, so the difference $\\infty - \\infty$ has no meaning.\n\nThe same formula is used when only one of $\\int f^+$ and $\\int f^-$ is finite, with value $\\pm \\infty$.\nThe integral is linear and monotone on integrable functions, and it satisfies $\\abs{\\int_X f \\, d\\mu} \\le \\int_X \\abs{f} \\, d\\mu$.\nOn a bounded interval, every Riemann integrable function is Lebesgue integrable and the two integrals agree.\nCommon notations are $\\int f \\, d\\mu$, $\\int f(x) \\, d\\mu(x)$ and $\\int f(x) \\, \\mu(dx)$.\n",
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