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  "body": "\n## Summary\n\nA vector measure is an additive set function whose values lie in a [[Banach space|06GZ956657]] instead of the real or complex numbers.\nThe basic notion is finite additivity; countable additivity is an extra property, with convergence of series in norm.\nVector measures arise from spectral theory, where a self-adjoint operator gives a projection-valued measure, and from integration of vector-valued functions.\nTheir theory, including the Bartle–Dunford–Schwartz integral and Lyapunov's convexity theorem, studies how far scalar measure theory extends to infinite dimensions.\n\n## Explanations\n\n### L1 human\n\nAn ordinary measure gives each set a number.\nA vector measure gives each set a vector, for example a function in some function space, and it adds up over disjoint sets.\nA simple example sends a set $E \\subseteq [0, 1]$ to its indicator function $\\mathbf{1}_E$, viewed as an element of $L^1([0, 1])$.\nNew phenomena appear in infinite dimensions: the same rule with values in $L^\\infty([0, 1])$ adds up over finitely many sets but not over countably many.\n",
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