Null set

In mathematical analysis, a null set N ⊂ R {displaystyle Nsubset mathbb {R} } is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory anticipates the development of Lebesgue measure since a null set necessarily has measure zero. More generally, on a given measure space M = ( X , Σ , μ ) {displaystyle M=(X,Sigma ,mu )} a null set is a set S ⊂ X {displaystyle Ssubset X} such that μ ( S ) = 0 {displaystyle mu (S)=0} . In mathematical analysis, a null set N ⊂ R {displaystyle Nsubset mathbb {R} } is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory anticipates the development of Lebesgue measure since a null set necessarily has measure zero. More generally, on a given measure space M = ( X , Σ , μ ) {displaystyle M=(X,Sigma ,mu )} a null set is a set S ⊂ X {displaystyle Ssubset X} such that μ ( S ) = 0 {displaystyle mu (S)=0} . Suppose A {displaystyle A} is a subset of the real line R {displaystyle mathbb {R} } such that where the Un are intervals and |U| is the length of U, then A 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 A for which the limit of the lengths of the covers is zero. Null sets include all finite sets, all countable sets, and even some uncountable sets such as the Cantor set. 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-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 N of R {displaystyle mathbb {R} } has null Lebesgue measure and is considered to be a null set in R {displaystyle mathbb {R} } if and only if: This condition can be generalised to R n {displaystyle mathbb {R} ^{n}} , using n-cubes instead of intervals. In fact, the idea can be made to make sense on any Riemannian manifold, even if there is no Lebesgue measure there.