Editing Lebesgue measure (section)

== Null sets ==
{{main|Null set}}
A subset of <math>\mathbb{R}^n</math> is a ''null set'' if, for every <math>\varepsilon > 0</math>, it can be covered with countably many products of ''n'' intervals whose total volume is at most <math>\varepsilon</math>. All [[countable]] sets are null sets.

If a subset of <math>\mathbb{R}^n</math> has [[Hausdorff dimension]] less than ''{{Mvar|n}}'' then it is a null set with respect to ''{{Mvar|n}}''-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the [[Euclidean metric]] on <math>\mathbb{R}^n</math> (or any metric [[Rudolf Lipschitz|Lipschitz]] equivalent to it). On the other hand, a set may have [[topological dimension]] less than {{Mvar|n}} and have positive ''{{Mvar|n}}''-dimensional Lebesgue measure. An example of this is the [[Smith–Volterra–Cantor set]] which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure.

In order to show that a given set ''<math display="inline">A</math>'' is Lebesgue-measurable, one usually tries to find a "nicer" set ''<math display="inline">B</math>'' which differs from ''<math display="inline">A</math>'' only by a null set (in the sense that the [[symmetric difference]] ''<math display="inline">(A \setminus B) \cup (B \setminus A)</math>'' is a null set) and then show that ''<math display="inline">B</math>'' can be generated using countable unions and intersections from open or closed sets.