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The term "null set" is sometimes also used to refer to the empty set; see that article. Alternatively, it may be used for any notion of negligible set; see that article. Wikipedia uses the term "null set" only in the measure theoretic sense.
Let X be a measurable space, let m be a measure on X, and let N be a measurable set in X. If m is a positive measure, then N is null if its measure m(N) is zero. If m is not a positive measure, then N is m-null if N is |m|-null, where |m| is the total variation of m; this is stronger than simply saying that m(N) = 0.
A nonmeasurable set is considered null if it is a subset of a null measurable set. Some references require a null set to be measurable; however, subsets of null sets are still negligible for measure-theoretic purposes.
When talking about null sets in Euclidean n-space Rn, it is usually understood that the measure being used is Lebesgue measure.
The empty set is always a null set. More generally, any countable union of null sets is null. Any measurable subset of a null set is itself a null set. Together, these facts show that the m-null sets of X form a sigma-ideal on X. Similarly, the measurable m-null sets form a sigma-ideal of the sigma-algebraIn mathematics, a sigma;-algebra (or sigma;-field X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S''. The concept is important in mathematical anal of measurable sets. Thus, null sets may be interpreted as negligible sets, defining a notion of almost everywhereIn measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. is a set with measure zero. If used for properties of the real numbers, t.
For Lebesgue measure on Rn, all 1-point setIn mathematics, a singleton is a set with exactly one element. For example, the set {0} is a singleton. Note that a set such as is also a singleton: the only element is a set (which itself is however not a singleton). A set is a singleton if and only if is are null, and therefore all countable sets are null. In particular, the set Q of rational numberIn mathematics, a rational number (or informally fraction is a ratio of two integers, usually written as the vulgar fraction a ''b where b is not zero. The set of all rational numbers is denoted by Q or in blackboard bold. Using the set-builder notation is is a null set, despite being dense in R. The Cantor setThe Cantor set introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. The Cantor set is defined by repeatedly removing the middle thirds of line segments. One starts by removing is an example of an uncountable null set in R.
More generally, a subset N of R is null if and only if:
This condition can be generalised to Rn, using n- cubes instead of intervals. In fact, the idea can be made to make sense on any topological manifold, even if there is no Lebesgue measure there.
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