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A measure in which all subsets of null sets are measurable is complete. Any non-complete measure can be completed to form a complete measure by asserting that subsets of null sets have measure zero. Lebesgue measure is an example of a complete measure; in some constructions, it is defined as the completion of a non-complete Borel measure.
In Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has zero measure. [16] In contrast, the set has a Hausdorff measure of 1 in its dimension of log 2 / log 3.
The Cantor set is an example of an uncountable set of Lebesgue measure 0 which is not of strong measure zero. [2] Borel's conjecture [1] 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 ...
The open interval (a, b) has the same measure, since the difference between the two sets consists only of the end points a and b, which each have measure zero. Any Cartesian product of intervals [ a , b ] and [ c , d ] is Lebesgue-measurable, and its Lebesgue measure is ( b − a )( d − c ) , the area of the corresponding rectangle .
Any set other than the empty set is called non-empty. In some textbooks and popularizations, the empty set is referred to as the "null set". [1] However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty).
In mathematics, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set of critical points) of a smooth function f from one Euclidean space or manifold to another is a null set, i.e., it has Lebesgue measure 0.
In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space ( X , Σ, μ ) is complete if and only if [ 1 ] [ 2 ]
Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than , although the measure cannot be exactly 1 (because otherwise the complement of its closure would be a nonempty open set with measure zero, which is impossible). [18]