Editing Cantor set (section)

=== Descriptive set theory ===
The Cantor set is a [[meagre set]] (or a set of first category) as a subset of <math>[0, 1]</math> (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 <math>\mathbb{Q}\cap[0,1]</math>, the Cantor set <math>\mathcal{C}</math> is "small" in the sense that it is a null set (a set of measure zero) and it is a meagre subset of <math>[0, 1]</math>.  However, unlike <math>\mathbb{Q}\cap[0,1]</math>, which is countable and has a "small" cardinality, <math>\aleph_0</math>, the cardinality of <math>\mathcal{C}</math> is the same as that of <math>[0, 1]</math>, the continuum <math>\mathfrak{c}</math>, and is "large" in the sense of cardinality.  In fact, it is also possible to construct a subset of <math>[0, 1]</math> that is meagre but of positive measure and a subset that is non-meagre but of measure zero:<ref>{{Cite book|title=Counterexamples in analysis|last=Gelbaum, Bernard R.|date=1964|publisher=Holden-Day|others=Olmsted, John M. H. (John Meigs Hubbell), 1911-1997|isbn=0486428753|location=San Francisco|oclc=527671}}</ref>  By taking the countable union of "fat" Cantor sets <math>\mathcal{C}^{(n)}</math> of measure <math>\lambda = (n-1)/n</math> (see Smith–Volterra–Cantor set below for the construction), we obtain a set <math display="inline">\mathcal{A} := \bigcup_{n=1}^{\infty}\mathcal{C}^{(n)}</math>which has a positive measure (equal to 1) but is meagre in [0,1], since each <math>\mathcal{C}^{(n)}</math> is nowhere dense.  Then consider the set <math display="inline">\mathcal{A}^{\mathrm{c}} = [0,1] \setminus\bigcup_{n=1}^\infty \mathcal{C}^{(n)}</math>.  Since <math>\mathcal{A}\cup\mathcal{A}^{\mathrm{c}} = [0,1]</math>, <math>\mathcal{A}^{\mathrm{c}}</math> cannot be meagre, but since <math>\mu(\mathcal{A})=1</math>, <math>\mathcal{A}^{\mathrm{c}}</math> must have measure zero.