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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]].\nA set $N \\subseteq X$ is a *null set*, or a $\\mu$-null set, if $N \\in \\Sigma$ and $\\mu(N) = 0$.\n\n## Remarks\n\nFor Lebesgue measure $\\lambda$ on $\\R$, every [[countable set|01TQB88H4C]] $\\set{x_1, x_2, \\ldots}$ is null.\nEach point has measure $0$, since $\\set{x} \\subseteq [x - \\varepsilon, x + \\varepsilon]$ for every $\\varepsilon \u003e 0$, and countable additivity gives total $0$.\nThe [[Cantor set|0BXNDZFQME]] is also a Lebesgue null set, though it is uncountable.\nThe interval $[0, 1]$ is not a null set, since $\\lambda([0, 1]) = 1$.\nA countable union of null sets is null, by countable subadditivity.\nSome authors call a set null when it is merely contained in a measurable set of measure $0$; the two notions agree when the measure is complete.\n",
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