The Carathéodory measure 
induced by 
, where the set function ![mu:S->[0,infty]](/images/equations/CaratheodoryMeasure/Inline3.svg)
is given, is the restriction
of the induced outer measure 
to the sigma-algebra 
of 
-measurable sets. Here
is a collection of subsets of a set 
, and the outer measure 
is induced by 
.
Perhaps somewhat surprisingly, even though 
is a measure induced by the
set function 
, it may not be the case that 
is an extension of 
. In the event that 
does extend 
, 
is called the Carathéodory
extension of 
.
