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  "body": "\n## Definition\n\nLet $\\Omega$ be a set and let $\\mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] on $\\Omega$.\nA probability measure on $(\\Omega, \\mathcal{F})$ is a function $P \\colon \\mathcal{F} \\to [0, 1]$ with these two properties.\n$P(\\Omega) = 1$.\nFor each sequence $A_1, A_2, \\dots$ of elements of $\\mathcal{F}$ with $A_i \\cap A_j = \\varnothing$ for all $i \\ne j$,\n$$P\\left(\\bigcup_{n=1}^{\\infty} A_n\\right) = \\sum_{n=1}^{\\infty} P(A_n).$$\n\n## Remarks\n\nThese are the [[probability axioms|0DAX8PMAJ4]]. In other words, a probability measure is a measure of total mass $1$.\nCountable additivity implies $P(\\varnothing) = 0$ and finite additivity: $P(A \\cup B) = P(A) + P(B)$ for disjoint $A$ and $B$.\nOn a finite set $\\Omega$ with the power set as $\\mathcal{F}$, each probability measure has the form $P(A) = \\sum_{\\omega \\in A} p(\\omega)$ with $p(\\omega) \\ge 0$ and $\\sum_{\\omega \\in \\Omega} p(\\omega) = 1$.\n\n## Explanations\n\n### L1 human\n\nA probability measure is a rule that gives each event a number between $0$ and $1$.\nThe whole sample space gets $1$, and the number of a union of separate pieces is the sum of the numbers of the pieces.\nFor a fair die, each face gets $1/6$, and the event \"even\" gets $3/6 = 1/2$.\n",
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