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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]].\nAn *atom* of $\\mu$ is a set $A \\in \\Sigma$ with $\\mu(A) \u003e 0$ such that for every $B \\in \\Sigma$ with $B \\subseteq A$, either $\\mu(B) = 0$ or $\\mu(A \\setminus B) = 0$.\nThe [[measure|PR11BZQQPG]] $\\mu$ is *non-atomic*, or *atomless*, if it has no atoms.\n\n## Remarks\n\nLebesgue measure $\\lambda$ on $\\R$ is non-atomic.\nIf $\\lambda(A) \u003e 0$, the function $t \\mapsto \\lambda(A \\cap (-\\infty, t])$ is continuous and increases from $0$ to $\\lambda(A)$.\nSo there is $t$ with $B = A \\cap (-\\infty, t]$ such that $\\lambda(B) \u003e 0$ and $\\lambda(A \\setminus B) \u003e 0$.\nThe [[Dirac measure|0ACRGYNHSR]] $\\delta_x$ is not non-atomic, since $\\set{x}$ is an atom: its only measurable subsets are $\\varnothing$ and $\\set{x}$.\nSome authors define an atom by the condition that every measurable $B \\subseteq A$ has $\\mu(B) = 0$ or $\\mu(B) = \\mu(A)$.\nFor sets $A$ of finite measure the two conditions agree.\n",
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