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  "body": "\n## Summary\n\nA signed measure is a countably additive set function on a σ-algebra that may take negative values.\nIt models quantities such as electric charge, which can be positive in some regions and negative in others.\nEvery signed measure is the difference of two ordinary [[measures|PR11BZQQPG]], one of them finite.\nSigned measures appear as differences of measures, as integrals $E \\mapsto \\int_E f \\, d\\mu$ of integrable functions, and as elements of dual spaces of function spaces.\n\n## Explanations\n\n### L1 human\n\nAn ordinary measure gives each set a size that is zero or positive.\nA signed measure also allows negative sizes, like a bank account in which some regions bring money and others cost money.\nIt must still add up correctly over countably many disjoint pieces.\nFor example, on $[-1, 1]$ the rule $\\nu(E) = \\int_E x \\, dx$ gives $\\nu([0, 1]) = \\tfrac12$ and $\\nu([-1, 0]) = -\\tfrac12$.\n",
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