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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]].\nLet $\\Sigma_\\mu$ be the family of sets $A \\cup S$, where $A \\in \\Sigma$ and $S$ is a subset of some [[null set|VQQM4Y550T]] $N \\in \\Sigma$.\nDefine $\\bar{\\mu} \\colon \\Sigma_\\mu \\to [0, \\infty]$ by $\\bar{\\mu}(A \\cup S) = \\mu(A)$.\nThe measure space $(X, \\Sigma_\\mu, \\bar{\\mu})$ is the *completion* of $(X, \\Sigma, \\mu)$.\n\n## Remarks\n\nThe family $\\Sigma_\\mu$ is a [[σ-algebra|04WSW8T3SY]] that contains $\\Sigma$.\nThe value $\\bar{\\mu}(A \\cup S)$ does not depend on the choice of $A$ and $S$.\nIndeed, if $A \\cup S = A' \\cup S'$ with $S \\subseteq N$ and $S' \\subseteq N'$, then $A$ and $A'$ differ only inside the null set $N \\cup N'$, so $\\mu(A) = \\mu(A')$.\nThen $\\bar{\\mu}$ is a [[complete measure|22K8CSQTYM]] that extends $\\mu$, and it is the smallest complete extension.\nFor example, let $X = \\set{1, 2, 3}$, $\\Sigma = \\set{\\varnothing, \\set{1, 2}, \\set{3}, X}$, and $\\mu(A) = 1$ if $3 \\in A$ and $0$ otherwise.\nThe null set $\\set{1, 2}$ gives $\\Sigma_\\mu$ equal to the power set of $X$, with $\\bar{\\mu}(A) = 1$ if $3 \\in A$ and $0$ otherwise.\nWith the same $\\Sigma$ and $\\nu(A) = 1$ if $1 \\in A$ and $0$ otherwise, the only null sets are $\\varnothing$ and $\\set{3}$.\nSo $\\Sigma_\\nu = \\Sigma$, and the power set is not the σ-algebra of the completion of $\\nu$.\nThe completion of Lebesgue measure on the Borel sets of $\\R$ is Lebesgue measure on the Lebesgue measurable sets.\n",
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