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 {In} of intervals such that N is contained in the union of the In and the total length of the In 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.

Read more about this topic:  Null Set

Famous quotes containing the word measure:

    Unless a group of workers know their work is under surveillance, that they are being rated as fairly as human beings, with the fallibility that goes with human judgment, can rate them, and that at least an attempt is made to measure their worth to an organization in relative terms, they are likely to sink back on length of service as the sole reason for retention and promotion.
    Mary Barnett Gilson (1877–?)