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  "body": "\n## Definition\n\nWork in a nonstandard extension of the real numbers that is countably saturated.\nLet $\\Omega$ be an internal set and let $\\mathcal{A}$ be an internal algebra of subsets of $\\Omega$.\nLet $\\mu \\colon \\mathcal{A} \\to {}^*[0, \\infty)$ be an internal [[finitely additive measure|KFWQ24D9M6]] such that $\\mu(\\Omega)$ is a finite hyperreal number.\nFor $A \\in \\mathcal{A}$ let $\\operatorname{st}\\mu(A)$ be the standard part of $\\mu(A)$, that is, the unique real number infinitely close to $\\mu(A)$.\nThe *Loeb measure* $L_\\mu$ of $\\mu$ is the unique [[measure|PR11BZQQPG]] on the σ-algebra $\\sigma(\\mathcal{A})$ generated by $\\mathcal{A}$ such that $L_\\mu(A) = \\operatorname{st}\\mu(A)$ for all $A \\in \\mathcal{A}$.\nThe [[measure space|P15VQS5DJ9]] $(\\Omega, \\sigma(\\mathcal{A}), L_\\mu)$ is the *Loeb space* of $\\mu$.\n\n## Remarks\n\nThe map $\\operatorname{st}\\mu$ is finitely additive on $\\mathcal{A}$.\nBy countable saturation, if a countable union of sets of $\\mathcal{A}$ lies in $\\mathcal{A}$, then it is already a finite union, so $\\operatorname{st}\\mu$ is countably additive on $\\mathcal{A}$.\nHence [[Carathéodory's extension theorem|Q8EXG7EKYW]] gives the extension, and it is unique because $\\operatorname{st}\\mu(\\Omega)$ is finite.\nMany authors call the [[completion|KJNSKV75VQ]] of this measure the Loeb measure.\n\nExample: let $N$ be an infinite hypernatural number, let $\\Omega = \\set{1, \\ldots, N}$, and let $\\mathcal{A}$ be the internal power set of $\\Omega$.\nLet $\\mu(A) = \\abs{A}/N$, where $\\abs{A}$ is the internal number of elements of $A$.\nThen $\\mu(\\Omega) = 1$ is finite, and $L_\\mu$ is a probability measure.\nThe standard part map $k \\mapsto \\operatorname{st}(k/N)$ pushes $L_\\mu$ forward to Lebesgue measure on $[0, 1]$.\n\nNon-example: on the same $\\Omega$, the internal counting measure $\\mu(A) = \\abs{A}$ has $\\mu(\\Omega) = N$, which is infinite.\nIt fails the finiteness clause, and the construction above does not apply directly.\nFor such $\\mu$ one can still define a Loeb measure, through an outer measure, but it need not be the unique extension.\n",
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