Measure zero
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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.
[edit] 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.
