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  "body": "\n## Definition\n\nLet $X$ be a set, let $\\mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] on $X$, and let $x \\in X$.\nThe Dirac measure at $x$ is the function $\\delta_x \\colon \\mathcal{F} \\to \\{0, 1\\}$ with\n$$\\delta_x(A) = \\begin{cases} 1 \u0026 \\text{if } x \\in A, \\\\ 0 \u0026 \\text{if } x \\notin A. \\end{cases}$$\n\n## Remarks\n\n$\\delta_x$ is a [[probability measure|0VGMEBVW1M]]. In terms of the [[indicator function|0VSB0XF4G7]], $\\delta_x(A) = \\mathbf{1}_A(x)$.\nThe integral of a measurable function $f$ with respect to $\\delta_x$ is $f(x)$.\nEach probability measure on a countable set $X$ with the power set as σ-algebra is a sum $\\sum_{x \\in X} p(x) \\, \\delta_x$ with weights $p(x) \\ge 0$ of sum $1$.\n\n## Explanations\n\n### L1 human\n\nThe Dirac measure puts all the probability on one point.\nIt describes an experiment whose result is certain, such as a coin with heads on both sides.\n",
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