Carathéodory measure

The measure induced by the outer Carathéodory measure , the latter being an outer measure defined on the class of all subsets of a metric space (with a metric ) such that

if . It was introduced by C. Carathéodory [1]. A set belongs to the domain of definition of , i.e. is -measurable, if and only if

for every (here ). If is -measurable, then . The domain of definition of a Carathéodory measure contains all Borel sets. If is an outer measure on the class of all subsets of a metric space such that every open set is -measurable, then is an outer Carathéodory measure.

References

[1]	C. Carathéodory, "Ueber das lineare Mass von Punktmengen, eine Verallgemeinerung des Längenbegriffs" Nachr. Gesell. Wiss. Göttingen (1914) pp. 404–426
[2]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
[3]	P.R. Halmos, "Measure theory" , v. Nostrand (1950)

Comments

An outer Carathéodory measure is also frequently called a metric outer measure, cf. [a1].

References

[a1]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)