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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]].\nThe measure $\\mu$, or the measure space, is *complete* if for every [[null set|VQQM4Y550T]] $N \\in \\Sigma$ and every subset $S \\subseteq N$, one has $S \\in \\Sigma$.\n\n## Remarks\n\nCounting measure on the power set of a set is complete, since every subset is measurable.\nLet $X = \\set{1, 2, 3}$, let $\\Sigma = \\set{\\varnothing, \\set{1, 2}, \\set{3}, X}$, and let $\\mu(A) = 1$ if $3 \\in A$ and $\\mu(A) = 0$ otherwise.\nThen $\\set{1, 2}$ is a null set, but its subset $\\set{1}$ is not in $\\Sigma$, so $\\mu$ is not complete.\nFor a complete measure, a subset of a null set is again a null set, by monotonicity.\nLebesgue measure on the σ-algebra of Lebesgue measurable sets is complete.\nIts restriction to the Borel sets is not complete: the Cantor set is a Borel null set with more subsets than there are Borel sets.\n",
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