Editing Null set (section)

==Examples==

Every finite or [[countably infinite]] subset of the [[real numbers]] {{tmath|\R}} is a null set. For example, the set of [[natural numbers]] {{tmath|\N}}, the set of [[rational numbers]] {{tmath|\Q}} and the set of [[algebraic numbers]] {{tmath|\mathbb A}} are all countably infinite and therefore are null sets when considered as subsets of the real numbers.

The [[Cantor set]] is an example of an uncountable null set. It is uncountable because it contains all real numbers between 0 and 1 whose [[Ternary numeral system|ternary]] expansion can be written using only 0’s and 2’s (see [[Cantor's diagonal argument]]), and it is null because it is constructed by beginning with the closed interval of real numbers from 0 to 1 and iteratively removing a third of the previous set, thereby multiplying the length by 2/3 with every step.