Cantor set measure theory pdf

Outer measures, premeasures, and product measures 179 chapter 2. Given 0 there exists a nlarge enough such that n n1 2n. The cantor set thus demonstrates that notions of size in terms of cardinality, measure, and baire category need not coincide. This distribution does not have a pdf, and it is totally mysterious from the masters level theory point of view. An introduction to elementary set theory guram bezhanishvili and eachan landreth 1 introduction in this project we will learn elementary set theory from the original historical sources by two key gures in the development of set theory, georg cantor 18451918 and richard dedekind 18311916. George cantor 18451918 was the originator of much of modern set theory.

Themainidea incalculatingthevolumeor measure ofasubset of rd consists of approximating this set by unions of other sets whose geometry is simple and whose volumes are known. Introduction pathological is an oft used word in the mathematical community, and in that context it has quite a di erent meaning than in everyday usage. Recall c is obtained from the closed interval 0,1 by. Cantor sets are the only disconnected, perfect, compact metric space up to a. Cantor s set is negligible one way to measure cantor s set by counting its elements shows that it is avery large set as large as the whole interval it is part of. In addition, cantor sets are uncountable, may have 0 or positive lebesgue measures, and are nowhere dense. Among his contributions to mathematics was the notion of the cantor set, which consists of points along a line segment, and possesses a number of fascinating properties. The cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. Outer measures, pre measures, and product measures 179 chapter 2. So all of the probability is concentrated on the cantor set c 1, which is what the measuretheoretic jargon calls a set of lebesgue measure zero, lebesgue measure being the measuretheoretic analog of ordinary length. The cantor set is a meager set or a set of first category as a subset of 0,1 although not as a subset of itself, since it is a baire space.

That is not a proof, you use the fact that the cantor set has a measure zero in order to prove that it has measure zero. Another way to measure it is by looking at theamount of space it occupieson the line. Fat cantor sets last month, i wrote about the cantor set, a mathematical space that is an interesting mix of small and large. Cantors set is negligible one way to measure cantors set by counting its elements shows that it is avery large set as large as the whole interval it is part of. With applications features a thorough introduction to cantor sets and applies these sets as a bridge between real analysis, probability, topology, and algebra the author fills a gap in the current literature by providing an introductory and integrated. Stat 8501 lecture notes baby measure theory charles j. A systematic and integrated approach to cantor sets and their applications to various branches of mathematics. He created set theory, which has become a fundamental theory in mathematics.

Themainidea incalculatingthevolumeormeasureofasubset of rd consists of approximating this set by unions of other sets whose geometry is simple and whose volumes are known. Cantor established the importance of onetoone correspondence between the members of two sets. A counterexample to this claim is the cantor set c. In r, we can use the euclidean distance to measure the length of the interval from 0 to 1, which has the length 1. Notice that the geometric series 1 n1 2n 1 3 converges to 1. Notes prepared by stanley burris march, 2001 set theory. Cantors work on set theory is linked to infinity and has implications for belief.

Set theory from cantor to cohen akihiro kanamori set theory is an autonomous and sophisticated. In mathematics, something is said to be \pathological if it is. Next, from the two remaining closed intervals we remove the open middle third i 2,1 19,29 and i 2,2 79,89, and so on. February 19 1845 january 6, 1918 was a german mathematician. Proof we will look at the pieces removed from the cantor set and the knowledge that m0. At a step, n, we have removed a total length n n1 2n 1 3 n. It is convenient to speakofvolumewhenreferringtosetsinrd. Measure of compact sets approximate from outside by opens outer and inner measures. The thirds set is a special case of smithvolterracantor sets in. The cantor set is an interesting example of an uncountable set of measure zero and has many interesting properties and consequences in the fields of set theory, topology, and fractal theory. View all references, some essential, some potential, and others actual. Cantor as we have seen, the naive use of classes, in particular the connection betweenconceptandextension,ledtocontradiction.