About: Strong measure zero set

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In mathematical analysis, a strong measure zero set is a subset A of the real line with the following property: for every sequence (εn) of positive reals there exists a sequence (In) of intervals such that |In| < εn for all n and A is contained in the union of the In. (Here |In| denotes the length of the interval In.) Every countable set is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every strong measure zero set has Lebesgue measure 0. The Cantor set is an example of an uncountable set of Lebesgue measure 0 which is not of strong measure zero.

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rdf:type	programming language
rdfs:label	Strong measure zero set (en)
rdfs:comment	In mathematical analysis, a strong measure zero set is a subset A of the real line with the following property: for every sequence (εn) of positive reals there exists a sequence (In) of intervals such that \|In\| < εn for all n and A is contained in the union of the In. (Here \|In\| denotes the length of the interval In.) Every countable set is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every strong measure zero set has Lebesgue measure 0. The Cantor set is an example of an uncountable set of Lebesgue measure 0 which is not of strong measure zero. (en)
dcterms:subject	Independence results Measure theory
Wikipage page ID	45200516 (xsd:integer)
Wikipage revision ID	1060138102 (xsd:integer)
Link from a Wikipage to another Wikipage	Independence (mathematical logic) Consistent Countable set Mathematical analysis Continuum hypothesis Independence results Émile Borel Meagre set Wacław Sierpiński Lebesgue measure Interval (mathematics) Measure theory ZFC Richard Laver Cantor set Real line Forcing (set theory)
sameAs	Strong measure zero set Strong measure zero set Strong measure zero set Strong measure zero set Strong measure zero set
dbp:wikiPageUsesTemplate	dbt:Reflist
has abstract	In mathematical analysis, a strong measure zero set is a subset A of the real line with the following property: for every sequence (εn) of positive reals there exists a sequence (In) of intervals such that \|In\| < εn for all n and A is contained in the union of the In. (Here \|In\| denotes the length of the interval In.) Every countable set is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every strong measure zero set has Lebesgue measure 0. The Cantor set is an example of an uncountable set of Lebesgue measure 0 which is not of strong measure zero. Borel's conjecture states that every strong measure zero set is countable. It is now known that this statement is independent of ZFC (the Zermelo–Fraenkel axioms of set theory, which is the standard axiom system assumed in mathematics). This means that Borel's conjecture can neither be proven nor disproven in ZFC (assuming ZFC is consistent).Sierpiński proved in 1928 that the continuum hypothesis (which is now also known to be independent of ZFC) implies the existence of uncountable strong measure zero sets. In 1976 Laver used a method of forcing to construct a model of ZFC in which Borel's conjecture holds. These two results together establish the independence of Borel's conjecture. The following characterization of strong measure zero sets was proved in 1973: A set A ⊆ R has strong measure zero if and only if A + M ≠ R for every meagre set M ⊆ R. This result establishes a connection to the notion of strongly meagre set, defined as follows: A set M ⊆ R is strongly meagre if and only if A + M ≠ R for every set A ⊆ R of Lebesgue measure zero. The dual Borel conjecture states that every strongly meagre set is countable. This statement is also independent of ZFC. (en)
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prov:wasDerivedFrom	wikipedia-en:Strong_measure_zero_set?oldid=1060138102&ns=0
page length (characters) of wiki page	4125 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Strong_measure_zero_set
is Link from a Wikipage to another Wikipage of	List of forcing notions Γ-space Émile Borel Richard Laver Borel conjecture List of statements independent of ZFC Rothberger space Selection principle Borel conjecture (set theory) Dual Borel conjecture Strongly meagre set
is Wikipage redirect of	Borel conjecture (set theory) Dual Borel conjecture Strongly meagre set
is foaf:primaryTopic of	wikipedia-en:Strong_measure_zero_set

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