To see that smz is much stronger than measure zero, consider for example the cantor set. But since the cantor set chas lebesgue measure zero, every subset of the cantor set is lebesgue measurable, i. The cantor set and the cantorlebesgue function note. Can this reasoning be extended to sets with uncountably many isola. For a presentation, i am learning about the cantor set and how it is homeomorphic to the padic numbers. 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. There is a way to construct zero lebesgue measure cantor sets that consists in successively removing gaps, that is, bounded open intervals, from an initial. One such example is a function whose deriva tive is bounded yet not riemann integrable, seemingly in violation of the.
Example 8 all countable subsets of r have lebesgue measure 0. A measure is a countably additive, nonnegative, extended real. Since a major focus of this text is lebesgue integration, it is also assumed that students have studied riemann integration in their. Lebesgue measure of an uncountable pointset mathematics. But since jcj jrj, it follows that pc pr, and hence pr jmj. Cantors set is obtained by successively removing intervals. Lebesgue measure of cantor set mathematics stack exchange. Lebesguemeasure on university of california, davis.
Cantor s set is negligible total length measure of intervals removed. We show that this special type of set is measurable set, borel set as well as borel measurable whose lebesgue measure is zero. A set is measurable if it belongs to a sigma algebra sof subsets of r. Incidentally, there is some sense in which the structure of the cantor set is fairly typical for closed sets. We show that this special type of set is measurable set, borel set. We use the cantorlebesgue function to show there are measurable sets which are not borel. The supplement to this section gives these results based on cardinality arguments but the supplement does not address the cantorlebesgue function.
A userfriendly introduction to lebesgue measure and. For a countable set, it is easy to construct a countable cover of arbitrary small intervals to show that the set has measure zero. Pdf lebesgue measure of generalized cantor set researchgate. In this paper, we discuss the construction and properties of generalized cantor set. The middle thirds cantor set is a subset of the interval c0 0,1. A sigma algebra sis a collections of subsets of r such that 1.
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