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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a measure space with $\\mu$ a [[probability measure|0VGMEBVW1M]], that is, $\\mu(X) = 1$.\nLet $-\\infty \\le a \u003c b \\le \\infty$ and let $g \\colon X \\to (a, b)$ be an [[integrable function|X7GTF3D98F]].\nLet $\\varphi \\colon (a, b) \\to \\R$ be a [[convex function|0VV0HSW1ZS]].\nThen $\\int_X g \\, d\\mu$ lies in $(a, b)$, and the negative part of $\\varphi \\circ g$ is integrable.\nMoreover,\n$$\\varphi\\Big( \\int_X g \\, d\\mu \\Big) \\le \\int_X \\varphi \\circ g \\, d\\mu,$$\nwhere the integrals are [[Lebesgue integrals|DWJF7ABPR1]] and the right side may be $+\\infty$.\n\n## Remarks\n\nProof idea: let $m = \\int g \\, d\\mu$.\nA convex function on an open interval has a supporting line at $m$: there is $c \\in \\R$ with $\\varphi(y) \\ge \\varphi(m) + c (y - m)$ for all $y \\in (a, b)$.\nPut $y = g(x)$ and integrate over $X$; the term $c(g - m)$ has integral $0$ because $\\mu(X) = 1$.\n\nExample: with $\\varphi(y) = y^2$ it gives $(\\int g \\, d\\mu)^2 \\le \\int g^2 \\, d\\mu$.\nIn probability this is $\\operatorname{Var}(Y) \\ge 0$.\nWith $\\varphi = \\exp$ it gives $\\exp(\\int g \\, d\\mu) \\le \\int e^{g} \\, d\\mu$.\n\nThe normalisation $\\mu(X) = 1$ is needed.\nFor $\\mu = 2 \\cdot$ Lebesgue measure on $[0, 1]$, $g = 1$ and $\\varphi(y) = y^2$, the left side is $4$ and the right side is $2$.\nWhen $\\mu$ is the probability measure with masses $w_i$ at finitely many points, the statement is the finite [[Jensen's inequality|0RWMRTPC87]].\n\n## Explanations\n\n### L1 human\n\nThe value of a convex function at an average is at most the average of its values.\nHere the average is an integral against a probability measure, so it covers both weighted sums and continuous averages.\nThe graph of a convex function bends upwards, which is why averaging after applying the function gives the larger result.\n\n### L2 human\n\nIn probability the inequality reads $\\varphi(\\mathbb{E}[Y]) \\le \\mathbb{E}[\\varphi(Y)]$ for an integrable random variable $Y$.\nIt yields the comparison $\\norm{f}_p \\le \\norm{f}_q$ for $p \\le q$ on probability spaces, the AM–GM inequality, and the non-negativity of the Kullback–Leibler divergence.\nThere is also a version for conditional expectations, used in martingale theory.\n",
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