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  "body": "\n## Summary\n\nA complex measure is a countably additive function from a σ-algebra to the complex numbers.\nIts real and imaginary parts are finite [[signed measures|VG2P7R91Z7]].\nComplex measures form a Banach space under the total variation norm.\nOn a locally compact Hausdorff space, the Riesz–Markov–Kakutani theorem identifies the complex Radon measures with the dual of the continuous functions vanishing at infinity.\nThey are the natural measures for Fourier analysis, where one integrates complex exponentials.\n\n## Explanations\n\n### L1 human\n\nA complex measure assigns a complex number to each measurable set, and adds up correctly over countably many disjoint pieces.\nOne can think of it as two signed measures side by side, one for the real part and one for the imaginary part.\nFor example, on $[0, 2\\pi]$ the rule $\\nu(E) = \\int_E e^{i t} \\, dt$ is a complex measure, and $\\nu([0, 2\\pi]) = 0$.\nUnlike a signed measure, a complex measure never takes infinite values.\n",
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