Let 
be a collection of subsets of a set 
, ![mu:S->[0,infty]](/images/equations/CaratheodoryMeasure/Inline3.svg)
a set function,
and 
the outer measure induced by 
. The measure 
that is the restriction of 
to the sigma-algebra 
of 
-measurable sets is called
the Carathéodory measure 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 
.
