Carathéodory's criterion

Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory that characterizes when a set is Lebesgue measurable.

Statement edit

Carathéodory's criterion: Let   denote the Lebesgue outer measure on   where   denotes the power set of   and let   Then   is Lebesgue measurable if and only if   for every   where   denotes the complement of   Notice that   is not required to be a measurable set.[1]

Generalization edit

The Carathéodory criterion is of considerable importance because, in contrast to Lebesgue's original formulation of measurability, which relies on certain topological properties of   this criterion readily generalizes to a characterization of measurability in abstract spaces. Indeed, in the generalization to abstract measures, this theorem is sometimes extended to a definition of measurability.[1] Thus, we have the following definition: If   is an outer measure on a set   where   denotes the power set of   then a subset   is called  –measurable or Carathéodory-measurable if for every   the equality

holds where   is the complement of  

The family of all  –measurable subsets is a σ-algebra (so for instance, the complement of a  –measurable set is  –measurable, and the same is true of countable intersections and unions of  –measurable sets) and the restriction of the outer measure   to this family is a measure.

See also edit

References edit

  1. ^ a b Pugh, Charles C. Real Mathematical Analysis (2nd ed.). Springer. p. 388. ISBN 978-3-319-17770-0.