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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\mathcal{C}$ be a family of subsets of $X$ with $\\emptyset \\in \\mathcal{C}$.\nA *fuzzy measure* on $(X, \\mathcal{C})$ is a map $g \\colon \\mathcal{C} \\to [0, \\infty]$ such that:\n\n1. $g(\\emptyset) = 0$.\n2. If $E, F \\in \\mathcal{C}$ and $E \\subseteq F$, then $g(E) \\le g(F)$.\n\nIf $X \\in \\mathcal{C}$ and $g(X) = 1$, the fuzzy measure is *normalised*.\n\n## Remarks\n\nOften $\\mathcal{C}$ is a $\\sigma$-algebra or the set of all subsets of a finite set $X$.\nExample: let $\\mu$ be a probability measure on a $\\sigma$-algebra $\\mathcal{C}$ and $g(E) = \\mu(E)^2$.\nThen $g(\\emptyset) = 0$, and $g$ is monotone because $\\mu$ is monotone and squaring is increasing on $[0, 1]$.\nIt is not additive: if $\\mu(E) = \\mu(F) = 1/2$ with $E \\cap F = \\emptyset$, then $g(E) + g(F) = 1/2$ but $g(E \\cup F) = 1$.\nNon-example: $g(E) = \\mu(E) - \\mu(E)^2$ vanishes on $\\emptyset$, but it is not monotone: $g(E) = 1/4$ for $\\mu(E) = 1/2$, while $g(X) = 0$.\nEvery [[measure|PR11BZQQPG]] is a fuzzy measure.\n",
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