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Concept
Null set
Summary
In mathematical analysis, a null set 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.
The notion of null set should not 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 a null set is a set such that
Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers and the set of rational numbers are both 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.
Suppose is a subset of the real line such that for every there exists a sequence of open intervals (where interval has length ) such that
then is a null set, also known as a set of zero-content.
In terminology of mathematical analysis, this definition requires that there be a sequence of open covers of for which the limit of the lengths of the covers is zero.
The empty set is always a null set. More generally, any countable union of null sets is null. Any subset of a null set is itself a null set. Together, these facts show that the -null sets of form a sigma-ideal on Similarly, the measurable -null sets form a sigma-ideal of the sigma-algebra of measurable sets. Thus, null sets may be interpreted as negligible sets, defining a notion of almost everywhere.
The Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space.
A subset of has null Lebesgue measure and is considered to be a null set in if and only if:
Given any positive number there is a sequence of intervals in such that is contained in the union of the and the total length of the union is less than
This condition can be generalised to using -cubes instead of intervals.
Official source
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