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  "body": "\n## Summary\n\nThe integral of a non-negative measurable function on a measure space is the supremum of the integrals of the simple functions that lie below it.\nIt always exists, as a number in $[0, \\infty]$.\nIt is the first step of the Lebesgue integral, and the monotone convergence theorem and Fatou's lemma are statements about it.\n\n## Explanations\n\n### L1 human\n\nFor a function that takes finitely many values, the integral is easy: add up each value times the measure of the set where it is taken.\nA general non-negative function is approached from below by such staircase functions.\nThe integral is the best value that these staircases reach.\nFor example, the function $f(x) = x$ on $[0, 1]$ has staircases below it with integrals as close to $1/2$ as one wants, and its integral is $1/2$.\nThe value $\\infty$ is allowed, as for the constant function $1$ on the whole real line.\n",
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