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  "body": "\n## Summary\n\nThe Bartle–Dunford–Schwartz integral integrates a scalar function against a countably additive [[vector measure|PS28BSAJKJ]] with values in a [[Banach space|06GZ956657]].\nThe integral of a function is then a vector of the Banach space.\nIt is built from simple functions, as the Lebesgue integral is.\nIt is a standard tool for spectral measures and for operators on spaces of continuous functions.\n\n## Explanations\n\n### L1 human\n\nAn ordinary measure gives each set a number, and the integral of a function is a weighted sum of these numbers.\nA vector measure gives each set a vector, so the weighted sum is a vector.\nFor a step function that takes the value $a_i$ on the set $E_i$, the integral is $\\sum_i a_i \\, \\mu(E_i)$.\nFor general functions one approximates them by step functions and takes the limit of the vectors.\n",
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