Measure zero - Definition and Overview

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.

Examples

Consider the Lebesgue measure μ over 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.

See also

• Almost everywhere
• Null set
• Cantor function
• Measure (mathematics)