Null set

Remarks

For Lebesgue measure $\lambda$ on $\R$, every countable set $\set{x_1, x_2, \ldots}$ is null. Each point has measure $0$, since $\set{x} \subseteq [x - \varepsilon, x + \varepsilon]$ for every $\varepsilon > 0$, and countable additivity gives total $0$. The Cantor set is also a Lebesgue null set, though it is uncountable. The interval $[0, 1]$ is not a null set, since $\lambda([0, 1]) = 1$. A countable union of null sets is null, by countable subadditivity. Some authors call a set null when it is merely contained in a measurable set of measure $0$; the two notions agree when the measure is complete.