In mathematical analysis, a null set in R {\displaystyle \mathbb {R} } is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. A null set is not to be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null. More generally, on a given measure space M = ( X , Σ , μ ) {\displaystyle M=(X,\Sigma ,\mu )} a null set is a set S ∈ Σ {\displaystyle S\in \Sigma } such that μ ( S ) = 0. {\displaystyle \mu (S)=0.}
Examples Every finite or countably infinite subset of the real numbers R {\displaystyle \mathbb {R} } is a null set. For example, the set of natural numbers N {\displaystyle \mathbb {N} } , the set of rational numbers Q {\displaystyle \mathbb {Q} } and the set of algebraic numbers A {\displaystyle \mathbb {A} } are all countably infinite and therefore are null sets when considered as subsets of the real numbers. The Cantor set is an example of an uncountable null set. It is uncountable because it contains all real numbers between 0 and 1 whose ternary expansion can be written using only 0s and 2s (see Cantor's diagonal argument), and it is null because it is constructed by beginning with the closed interval of real numbers from 0 to 1 and iteratively removing a third of the previous set, thereby multiplying the length by 2/3 with every step. The set of Liouville numbers is another example of an uncountable null set.
Definition for Lebesgue measure The Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. A subset N {\displaystyle N} of the real line R {\displaystyle \mathbb {R} } has null Lebesgue measure and is considered to be a null set (also known as a set of zero-content) in R {\displaystyle \mathbb {R} } if and only if:
(In terminology of mathematical analysis, this definition requires that there be a sequence of open covers of A {\displaystyle A} for which the limit of the lengths of the covers is zero.) This condition can be generalised to R n , {\displaystyle \mathbb {R} ^{n},} using n {\displaystyle n} -cubes instead of intervals. In fact, the idea can be made to make sense on any manifold, even if there is no Lebesgue measure there. For instance:
With respect to R n , {\displaystyle \mathbb {R} ^{n},} all singleton sets are null, and therefore all countable sets are null. In particular, the set Q {\displaystyle \mathbb {Q} } of rational numbers is a null set, despite being dense in R . {\displaystyle \mathbb {R} .}
The standard construction of the Cantor set is an example of a null uncountable set in R ; {\displaystyle \mathbb {R} ;} however other constructions are possible which assign the Cantor set any measure whatsoever. All the subsets of R n {\displaystyle \mathbb {R} ^{n}} whose dimension is smaller than n {\displaystyle n} have null Lebesgue measure in R n . {\displaystyle \mathbb {R} ^{n}.} For instance straight lines or circles are null sets in R 2 . {\displaystyle \mathbb {R} ^{2}.}
Sard's lemma: the set of critical values of a smooth function has measure zero. If λ {\displaystyle \lambda } is Lebesgue measure for R {\displaystyle \mathbb {R} } and π is Lebesgue measure for R 2 {\displaystyle \mathbb {R} ^{2}} , then the product measure λ × λ = π . {\displaystyle \lambda \times \lambda =\pi .} In terms of null sets, the following equivalence has been styled a Fubini's theorem:
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