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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]] and let $E$ be a [[Banach space|0BDTTM24DP]] over $\\R$ or $\\C$.\nAn *integrable simple function* is a function $s = \\sum_{i=1}^{n} \\mathbf{1}_{A_i} e_i$ with $e_i \\in E$ and pairwise disjoint $A_i \\in \\Sigma$ with $\\mu(A_i) \u003c \\infty$.\nIts integral is $\\int_X s \\, d\\mu = \\sum_{i=1}^{n} \\mu(A_i) \\, e_i$.\nA function $f \\colon X \\to E$ is *Bochner integrable* if both of the following hold.\n\n1. The function $f$ is [[strongly measurable|JDM2M1FRY6]].\n2. There are integrable simple functions $s_1, s_2, \\dots$ with $\\lim_{n \\to \\infty} \\int_X \\norm{f - s_n} \\, d\\mu = 0$.\n\nThe integrals in clause 2 are [[integrals of non-negative functions|DG76G5Z6XW]].\nThe *Bochner integral* of $f$ is then $\\int_X f \\, d\\mu = \\lim_{n \\to \\infty} \\int_X s_n \\, d\\mu$, a limit in $E$.\n\n## Remarks\n\nThe limit exists, since $\\norm{\\int s_n - \\int s_m} \\le \\int \\norm{s_n - s_m}$, so the integrals form a Cauchy sequence in $E$.\nIt does not depend on the choice of the $s_n$.\n\nExample: let $X = [0, 1]$ with Lebesgue measure and let $f \\colon X \\to E$ be continuous.\nThe step functions $s_n$ equal to $f(k/n)$ on $[k/n, (k+1)/n)$ for $0 \\le k \u003c n$ and to $f(1)$ at $1$ converge to $f$ uniformly, so $\\int \\norm{f - s_n} \\to 0$.\nSo $f$ is Bochner integrable, and its integral is the limit of the Riemann sums $\\frac{1}{n} \\sum_{k=0}^{n-1} f(k/n)$.\n\nNon-example: take $X = \\R$ with Lebesgue measure, $E = \\R$ and $f = 1$.\nIt is strongly measurable, but every integrable simple function vanishes outside a set of finite measure, so $\\int \\norm{f - s} = \\infty$ for each of them.\n\nBochner's theorem: a strongly measurable function $f$ is Bochner integrable if and only if $\\int_X \\norm{f} \\, d\\mu \u003c \\infty$.\nFor $E = \\R$ or $E = \\C$ the Bochner integral is the Lebesgue integral.\nFor a bounded linear map $T \\colon E \\to F$ one has $T \\int f \\, d\\mu = \\int T f \\, d\\mu$.\n",
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