Cantor set
Descriptive set theory

The Cantor set is a meagre set (or a set of first category) as a subset of (although not as a subset of itself, since it is a Baire space). The Cantor set thus demonstrates that notions of "size" in terms of cardinality, measure, and (Baire) category need not coincide. Like the set Q ∩ {\displaystyle \mathbb {Q} \cap }, the Cantor set C {\displaystyle {\mathcal {C}}} is "small" in the sense that it is a null set (a set of measure zero) and it is a meagre subset of. However, unlike Q ∩ {\displaystyle \mathbb {Q} \cap }, which is countable and has a "small" cardinality, ℵ 0 {\displaystyle \aleph _{0}}, the cardinality of C {\displaystyle {\mathcal {C}}} is the same as that of, the continuum c {\displaystyle {\mathfrak {c}}}, and is "large" in the sense of cardinality. In fact, it is also possible to construct a subset of that is meagre but of positive measure and a subset that is non-meagre but of measure zero: By taking the countable union of "fat" Cantor sets C (n) {\displaystyle {\mathcal {C}}^{(n)}} of measure λ = (n − 1) / n {\displaystyle \lambda =(n-1)/n} (see Smith–Volterra–Cantor set below for the construction), we obtain a set A:= ⋃ n = 1 ∞ C (n) {\textstyle {\mathcal {A}}:=\bigcup _{n=1}^{\infty }{\mathcal {C}}^{(n)}} which has a positive measure (equal to 1) but is meagre in, since each C (n) {\displaystyle {\mathcal {C}}^{(n)}} is nowhere dense. Then consider the set A c = ∖ ⋃ n = 1 ∞ C (n) {\textstyle {\mathcal {A}}^{\mathrm {c} }=\setminus \bigcup _{n=1}^{\infty }{\mathcal {C}}^{(n)}}. Since A ∪ A c = {\displaystyle {\mathcal {A}}\cup {\mathcal {A}}^{\mathrm {c} }=}, A c {\displaystyle {\mathcal {A}}^{\mathrm {c} }} cannot be meagre, but since μ (A) = 1 {\displaystyle \mu ({\mathcal {A}})=1}, A c {\displaystyle {\mathcal {A}}^{\mathrm {c} }} must have measure zero.
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