Null Set - Lebesgue Measure

Lebesgue Measure

The Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space.

A subset N of R has null Lebesgue measure and is considered to be a null set in R if and only if:

Given any positive number ε, there is a sequence {I_n} of intervals such that N is contained in the union of the I_n and the total length of the I_n is less than ε.

This condition can be generalised to Rn, using n-cubes instead of intervals. In fact, the idea can be made to make sense on any topological manifold, even if there is no Lebesgue measure there.

For instance:

• With respect to Rn, all 1-point sets are null, and therefore all countable sets are null. In particular, the set Q of rational numbers is a null set, despite being dense in R.
• The standard construction of the Cantor set is an example of a null uncountable set in R; however other constructions are possible which assign the Cantor set any measure whatsoever.
• All the subsets of Rn whose dimension is smaller than n have null Lebesgue measure in Rn. For instance straight lines or circles are null sets in R2.
• Sard's lemma: the set of critical values of a smooth function has measure zero.