Example of uncountable set with lebesgue measure zero
Examples of sets of measure 0 A nite, A x 1, So the Cantor set is uncountable. Sets of Measure Zero. To see that it has measure 0, notice that A n [2n k1 B n, k, B n, k [a so A has measure 0. Sets of Measure Zero. To see that it has measure 0, notice that A n [2n k1 B n, k, BDe nition 1. A subset Z R is a zero set if for any 0 there exists a countable collection f(a i; b i)gof intervals which cover Zsuch that X1 i1 (b i a i): This sum is called the total length of the collection f(a i; b i)g. We will also say that a zero set has measure zero. Let us wrap our heads around this de nition. example of uncountable set with lebesgue measure zero
The Cantor set is indeed the canonical example here. There is an interesting question of just how different from the Cantor set you can get and still have this property. You can find a discussion of this issue at Examples of uncountable sets with zero Lebesgue measure.
However, there are Lebesguemeasurable sets which are not Borel sets. Any countable set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of rational numbers is 0, although the set is dense in R. The Cantor set is an example of an uncountable set that has Lebesgue measure zero. Stack Exchange network consists of 174 Q& A communities including Stack Overflow, the largest, most trusted online community for developers to learn, shareexample of uncountable set with lebesgue measure zero lebesgue measure and countable sets [closed [0, 1. why is that, i know the measure of a countable set is zero, but why i cant find an explanation for this, but how can a set with cardinality of \infty still be zero Do sets with positive Lebesgue measure have same cardinality as R? 0. Question on measure theory. 5.