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  "body": "\n## Summary\n\nA Loeb measure is a genuine countably additive [[measure|PR11BZQQPG]] built from a finitely additive measure inside a nonstandard model of analysis.\nOne takes the standard part of the nonstandard measure and extends it to a σ-algebra.\nIntroduced by Peter Loeb, it lets hyperfinite objects, such as a finite set with an infinite number of points, serve as ordinary measure spaces.\nIt is the main tool of nonstandard measure theory and gives nonstandard constructions of Lebesgue measure and of Brownian motion.\n\n## Explanations\n\n### L1 human\n\nIn nonstandard analysis there are \"finite\" sets whose number of elements $N$ is an infinitely large hypernatural number.\nOn such a set one can give each point the infinitesimal weight $1/N$, as for a fair die with $N$ faces.\nThe resulting sizes are hyperreal numbers, so one rounds each of them to the nearest real number.\nLoeb showed that the rounded sizes extend to a true probability measure.\nFor example, mapping the points $k/N$ for $k = 1, \\ldots, N$ to their standard parts in $[0, 1]$ turns this uniform measure into Lebesgue measure on $[0, 1]$.\n",
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