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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]] whose measure is [[complete|22K8CSQTYM]].\nA *model space* is built from a number $a \\in [0,1]$ and a finite or countable family of numbers $p_k \u003e 0$ with $a + \\sum_k p_k = 1$.\nIt is the disjoint union of $[0, a]$ with points $x_k$.\nLebesgue measurable subsets of $[0,a]$ get their [[Lebesgue measure|2Z2QVWZ262]], and each point $x_k$ gets mass $p_k$.\nTwo complete probability spaces are *isomorphic mod $0$* if a [[null set|VQQM4Y550T]] can be removed from each so that the following holds.\nThere is a bijection between the remaining sets that is measurable in both directions and preserves the measure.\nThe space $(\\Omega, \\mathcal{F}, P)$ is *standard* if it is isomorphic mod $0$ to a model space.\n\n## Remarks\n\nExample: the unit square $[0,1]^2$ with completed Lebesgue measure is standard with $a = 1$ and no atoms.\nRemove the null set of points with a dyadic rational coordinate, and interleave the binary digits of the two coordinates.\nThis gives a bijection mod $0$ with $[0,1]$ that sends dyadic squares to dyadic intervals of the same measure, so it preserves the measure.\nNon-example: let $Z \\subseteq [0,1]$ have Lebesgue inner measure $0$ and outer measure $1$, and let $\\Omega = [0,1] \\setminus Z$ with $P(A \\cap \\Omega) = \\lambda(A)$ for Lebesgue measurable $A$.\nThis \"perforated interval\" is a complete probability space that is known not to be standard.\nRokhlin's Lebesgue spaces are the same spaces, defined through a countable separating family of sets and a completeness condition.\n",
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