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  "body": "\n## Definition\n\nLet $(X, \\Sigma)$ be a [[measurable space|N4KDT83V3S]] and let $\\mu$ and $\\nu$ be [[measures|PR11BZQQPG]] on $\\Sigma$.\nThe measures $\\mu$ and $\\nu$ are *mutually singular* if there are sets $A, B \\in \\Sigma$ such that:\n\n1. $A \\cap B = \\emptyset$ and $A \\cup B = X$.\n2. $\\mu(B) = 0$.\n3. $\\nu(A) = 0$.\n\nThis is written $\\mu \\perp \\nu$.\n\n## Remarks\n\nExample: on the Borel sets of $\\R$, let $\\lambda$ be Lebesgue measure and let $\\delta_0$ be the Dirac measure at $0$.\nTake $A = \\R \\setminus \\{0\\}$ and $B = \\{0\\}$.\nThen $\\lambda(B) = 0$ and $\\delta_0(A) = 0$, so $\\lambda \\perp \\delta_0$.\n\nNon-example: let $\\mu$ be Lebesgue measure restricted to $[0, 1]$ and $\\nu$ Lebesgue measure restricted to $[0, 2]$.\nIf $\\mu(B) = 0$, then $B \\cap [0, 1]$ has length $0$, so $A$ contains almost all of $[0, 1]$ and $\\nu(A) \\ge 1$.\nSo clause 3 fails for every choice, and $\\mu$ and $\\nu$ are not mutually singular.\n\nThe relation is symmetric.\nFor signed or complex measures, $\\mu \\perp \\nu$ means that the total variations $\\abs{\\mu}$ and $\\abs{\\nu}$ are mutually singular.\nA measure $\\nu$ is called singular with respect to $\\mu$ when $\\nu \\perp \\mu$.\n",
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