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The Cantor set can be seen as the [[compact group]] of binary sequences, and as such, it is endowed with a natural [[Haar measure]]. When normalized so that the measure of the set is 1, it is a model of an infinite sequence of coin tosses. Furthermore, one can show that the usual [[Lebesgue measure]] on the interval is an image of the Haar measure on the Cantor set, while the natural injection into the ternary set is a canonical example of a [[singular measure]]. It can also be shown that the Haar measure is an image of any [[probability]], making the Cantor set a universal probability space in some ways.
In [[Lebesgue measure]] theory, the Cantor set is an example of a set which is uncountable and has zero measure.<ref>{{cite web | url=http://theoremoftheweek.wordpress.com/2010/09/30/theorem-36-the-cantor-set-is-an-uncountable-set-with-zero-measure/ | title=Theorem 36: the Cantor set is an uncountable set with zero measure | first=Laura | last=Irvine | website=Theorem of the week | access-date=2012-09-27 | archive-url=https://web.archive.org/web/20160315212203/https://theoremoftheweek.wordpress.com/2010/09/30/theorem-36-the-cantor-set-is-an-uncountable-set-with-zero-measure/ | archive-date=2016-03-15 | url-status=dead }}</ref> In contrast, the set has a [[Hausdorff measure]] of 1 in its dimension of log 2 / log 3.<ref>
{{cite book |last=Falconer |first=K. J. |date=July 24, 1986 |title=The Geometry of Fractal Sets |url=http://mate.dm.uba.ar/~umolter/materias/referencias/1.pdf |pages=14–15 |publisher=Cambridge University Press |isbn=9780521337052}}
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===Cantor numbers===
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