The Carathéodory extension theorem states that a premeasure 
on an algebra of sets 
over 
extends to a measure 
on the sigma-algebra 
generated by 
such that 
for every 
. If 
is sigma-finite, then this Carathéodory
extension is unique.
The construction begins with the outer measure
 |
for 
.
A set 
is declared Carathéodory measurable when
 |
for every 
. The Carathéodory measurable sets form a
sigma-algebra containing 
, and the restriction of 
to them is a complete measure.
Its restriction to 
is the extension asserted by the theorem.
The theorem is the standard mechanism for constructing measures from values first specified on simpler families of sets. For example, it underlies the construction of Lebesgue measure from interval lengths.
