Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory. Its statement is as follows: Let
λ
∗
denote the Lebesgue outer measure on
R
n
, and let
E
⊆
R
n
. Then
E
is Lebesgue measurable if and only if
λ
∗
(
A
)
=
λ
∗
(
A
∩
E
)
+
λ
∗
(
A
∩
E
c
)
for every
A
⊆
R
n
. Notice that
A
is not required to be a measurable set.
The Carathéodory criterion is of considerable importance because, in contrast to Lebesgue's original formulation of measurability, which relies on certain topological properties of
R
, 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. Thus, we have the following definition: Let
μ
∗
be an outer measure on a set
X
. Then
E
⊂
X
is called
μ
∗
–measurable if for every
A
⊂
X
, the equality
μ
∗
(
A
)
=
μ
∗
(
A
∩
E
)
+
μ
∗
(
A
∩
E
c
)
holds.
