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  "body": "\n## Statement\n\nLet $\\mathfrak{A}$ be a [[Boolean algebra|0XADJKM699]] in which every countable subset has a supremum, and let $m \\colon \\mathfrak{A} \\to [0, 1]$.\nThe pair $(\\mathfrak{A}, m)$ is a *probability measure algebra* if three conditions hold.\n\n1. $m(1) = 1$, and $m(a) = 0$ only for $a = 0$.\n2. $m\\left(\\bigvee_k a_k\\right) = \\sum_k m(a_k)$ for every sequence of pairwise disjoint elements $a_k$.\n3. The set $\\mathfrak{A}$ is separable for the metric $d(a, b) = m(a \\mathbin{\\triangle} b)$.\n\nAssume also that $(\\mathfrak{A}, m)$ is atomless: every $a \\ne 0$ has some $b \\le a$ with $0 \u003c m(b) \u003c m(a)$.\nLet $\\mathfrak{L}$ be the algebra of [[Lebesgue measurable|2Z2QVWZ262]] subsets of $[0,1]$ modulo [[null sets|VQQM4Y550T]], with $\\lambda$ induced by Lebesgue measure.\nThen there is a Boolean algebra isomorphism $\\Phi \\colon \\mathfrak{A} \\to \\mathfrak{L}$ with $\\lambda(\\Phi(a)) = m(a)$ for all $a \\in \\mathfrak{A}$.\n\n## Remarks\n\nIdea of proof: take a dense sequence in $\\mathfrak{A}$ and the finite partitions of $1$ that it generates.\nSince the algebra is atomless, $m$ takes all values between $0$ and $m(a)$ on elements below $a$.\nThis lets one send each partition to a partition of $[0,1]$ into finite unions of intervals with the same measures, compatibly with refinement.\nThe map then extends by continuity in the metric $d$.\nWithout atomlessness the result fails: the algebra with two atoms of measure $1/2$ has four elements, while $\\mathfrak{L}$ is infinite.\nWithout separability it fails as well: the measure algebra of the product measure on $\\set{0,1}^I$ with $I$ uncountable is atomless but not separable.\nAs a consequence, the measure algebra of every atomless [[standard probability space|QDDRDW9PXS]] is isomorphic to that of $[0,1]$.\n\n## Explanations\n\n### L1 human\n\nIgnore sets of measure zero.\nThen there is essentially only one way to spread probability evenly with no lumps, as long as the space is not too large.\nEvery such system of events looks like the system of measurable subsets of the unit interval with length as probability.\nThis is why the unit interval can serve as a model for many random experiments.\n\n### L2 human\n\nThe theorem lets ergodic theory move between point maps and automorphisms of measure algebras.\nTogether with the theory of standard probability spaces, it shows that many probability spaces of analysis are isomorphic to $[0,1]$ up to null sets.\nAn example is $[0,1]^\\N$ with the product of Lebesgue measures.\nMaharam's theorem classifies the non-separable case.\n",
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