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  "body": "\n## Definition\n\nLet $X$ be a finite nonempty set, let $\\mathcal{R}$ be a [[family of subsets|0C21Q694AG]] of $X$, called ranges, and let $0 \u003c \\varepsilon \\le 1$.\nA set $N \\subseteq X$ is an *$\\varepsilon$-net* of $(X, \\mathcal{R})$ if every range $R \\in \\mathcal{R}$ with $\\abs{R} \\ge \\varepsilon \\abs{X}$ satisfies $R \\cap N \\ne \\emptyset$.\n\n## Remarks\n\nExample: let $X = \\set{1, 2, \\dots, 100}$, let $\\mathcal{R}$ be the sets of the form $\\set{a, a+1, \\dots, b}$, and let $\\varepsilon = 1/10$.\nThe set $N = \\set{10, 20, \\dots, 100}$ is an $\\varepsilon$-net.\nIndeed, a range with at least $10$ elements consists of at least $10$ consecutive integers, so it contains a multiple of $10$.\n\nNon-example: for the same $X$ and $\\mathcal{R}$, the set $N' = \\set{20, 40, \\dots, 100}$ is not a $1/10$-net.\nThe range $\\set{1, \\dots, 19}$ has $19 \\ge 10$ elements but misses $N'$.\n\nThe same notion is used for a probability measure $\\mu$ on a set $X$: then $N$ is an $\\varepsilon$-net if it meets every range $R$ with $\\mu(R) \\ge \\varepsilon$.\nThe case above is $\\mu$ equal to the uniform measure on a finite $X$.\n",
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