Null set

A null set in a measure space $(X,\Sigma,\mu)$ is a measurable set $N\in\Sigma$ such that $\mu(N)=0$.

Null sets are negligible for many purposes: statements that fail only on a null set are said to hold almost everywhere . Many constructions treat functions that differ only on a null set as essentially the same.

Examples:

• The empty set is always a null set.
• In $\mathbb R$ with Lebesgue measure , any singleton $\{x\}$ is a null set.
• In $\mathbb R$ with Lebesgue measure, every countable subset of $\mathbb R$ is a null set.