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  "body": "\n## Definition\n\nLet $(\\Omega, \\Sigma)$ be a measurable space, let $E$ be a real [[Banach space|06GZ956657]], and let $\\mu \\colon \\Sigma \\to E$ be a countably additive [[vector measure|PS28BSAJKJ]].\nA set $N \\in \\Sigma$ is $\\mu$-null if $\\mu(A) = 0$ for every $A \\in \\Sigma$ with $A \\subseteq N$.\n\n1. For a [[simple function|7WMBX7M4P6]] $s = \\sum_{i=1}^{k} a_i \\mathbf{1}_{A_i}$ with $a_i \\in \\R$ and $A_i \\in \\Sigma$, and for $B \\in \\Sigma$, put $\\int_B s \\, d\\mu = \\sum_{i=1}^{k} a_i \\, \\mu(A_i \\cap B)$.\n2. A [[measurable|YYNCJSR0PJ]] $f \\colon \\Omega \\to \\R$ is $\\mu$-integrable if there are simple functions $s_n \\to f$, pointwise outside a $\\mu$-null set, such that $\\int_B s_n \\, d\\mu$ converges in $E$ for every $B \\in \\Sigma$.\n3. For such $f$, the *Bartle–Dunford–Schwartz integral* is $\\int_B f \\, d\\mu = \\lim_{n \\to \\infty} \\int_B s_n \\, d\\mu$ for $B \\in \\Sigma$.\n\nThe limit in clause 3 does not depend on the choice of the sequence $(s_n)$.\n\n## Remarks\n\nExample: every bounded measurable $f$ is $\\mu$-integrable.\nTake simple $s_n$ with $\\abs{s_n - f} \\le 1/n$ everywhere.\nFor a simple $s$ with $\\abs{s} \\le c$ one has $\\norm{\\int_B s \\, d\\mu} \\le c \\, \\norm{\\mu}(\\Omega)$.\nHere $\\norm{\\mu}(\\Omega)$ is the supremum of $\\norm{\\sum_i a_i \\mu(A_i)}$ over finite measurable partitions $(A_i)$ of $\\Omega$ and reals $\\abs{a_i} \\le 1$; it is finite when $\\mu$ is countably additive.\nApply this to $s = s_n - s_m$.\nSo $\\int_B s_n \\, d\\mu$ is a Cauchy sequence in $E$.\n\nNon-example: let $\\mu$ be Lebesgue measure on $(0, 1]$, viewed as a vector measure with values in $\\R$, and $f(x) = 1/x$.\nFor a real-valued finite measure, integrability in the sense of clause 2 is the same as Lebesgue integrability.\nSince $\\int_0^1 dx / x = \\infty$, no sequence as in clause 2 exists, and $f$ is not $\\mu$-integrable.\n\nWhen $E = \\R$ and $\\mu$ is a finite signed measure, the integral is the Lebesgue integral against $\\mu$.\nFor each continuous linear functional $x^*$ on $E$, $x^* \\big( \\int_B f \\, d\\mu \\big) = \\int_B f \\, d(x^* \\circ \\mu)$.\n",
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