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.
Carathéodory's criterion: Let
denote the Lebesgue outer measure on
denotes the power set of
is Lebesgue measurable if and only if
λ
λ
λ
denotes the complement of
is not required to be a measurable set.
[1] 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
denotes the power set of
then a subset
is called
–measurable or Carathéodory-measurable if for every
the equality
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.
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