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  "body": "\n## Example\n\nStart with $C_0 = [0, 1]$.\nAt step $n \\ge 1$, the set $C_{n-1}$ is a union of $2^{n-1}$ disjoint closed intervals.\nRemove from the middle of each of them an open interval of length $4^{-n}$, and call the result $C_n$.\nThe Smith–Volterra–Cantor set is $S = \\bigcap_{n \\ge 0} C_n$.\nThen $S$ is a [[closed set|0ZCK8DQYN1]], it is [[nowhere dense|0VQYCCQVJ5]] in $\\R$, and its [[Lebesgue measure|2Z2QVWZ262]] is $1/2$.\n\n## Remarks\n\nStep $1$ removes $(3/8, 5/8)$ and leaves $[0, 3/8] \\cup [5/8, 1]$.\nBy induction, each of the $2^n$ intervals of $C_n$ has length $2^{-n-1} + 2^{-2n-1}$.\nThis is more than $4^{-n-1}$, so each removal is possible.\nThe total length removed is\n$$\\sum_{n=1}^{\\infty} 2^{n-1} \\cdot 4^{-n} = \\frac{1}{4} \\sum_{n=0}^{\\infty} 2^{-n} = \\frac{1}{2},$$\nso $\\lambda(S) = 1 - 1/2 = 1/2$.\nThe set $S$ is closed as an intersection of closed sets.\nThese lengths tend to $0$, so $S$ contains no interval; a closed set with empty interior is nowhere dense.\nSo a nowhere dense set can have positive measure, unlike the middle-thirds [[Cantor set|0BXNDZFQME]], which has measure $0$.\nThe indicator function of $S$ is bounded and discontinuous at every point of $S$, a set of positive measure, so it is not Riemann integrable.\n\n## Explanations\n\n### L1 human\n\nThe ordinary Cantor set removes a third of each interval at every step and ends with total length $0$.\nThe Smith–Volterra–Cantor set removes much smaller gaps, of length $1/4$, then $1/16$, then $1/64$, and so on.\nThe gaps still reach everywhere, so the set contains no interval at all.\nYet the gaps only add up to length $1/2$, so the remaining dust of points still has total length $1/2$.\n",
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