MEASURE ZERO

Let μ be a measure on a sigma algebra Σ of subsets of a set ''X''. An element ''A'' in Σ is said to have 'measure zero' if μ(''A'')=0.
Any set of measure zero is a null set. The opposite is not true, because a null set is not required to be measurable, that is, to be an element in Σ. However, any null set is a subset of a set of measure zero. If the measure space is complete, then a set is null if and only if it has measure zero.

Contents

Examples

See also

Examples

Consider Lebesgue measure on the real numbers.

★ Any countable set of real numbers has measure zero. In particular, the set of all rational numbers and the set of discontinuities of a monotonic function have measure zero.

★ An ''uncountable'' set of real numbers which has measure zero is the Cantor set.

★ Sard's lemma: the set of critical values of a smooth function has measure zero.

See also

★ Almost everywhere

★ Null set

★ Cantor function

★ Measure (mathematics)