Given a set 
, a set function ![mu^*:2^X->[0,infty]](http://mathworld.wolfram.com/images/equations/OuterMeasure/Inline2.svg)
is said to be an outer measure provided that 
and that 
is countably monotone, where 
is the empty set.
Given a collection 
of subsets of 
and an arbitrary set function ![mu:S->[0infty]](http://mathworld.wolfram.com/images/equations/OuterMeasure/Inline8.svg)
, one can define a new set function 
by setting 
and defining, for each non-empty subset 
,
 |
where the infimum is taken over all countable collections 
of sets in 
which cover 
. The resulting function ![mu^*:2^X->[0infty]](http://mathworld.wolfram.com/images/equations/OuterMeasure/Inline15.svg)
is an outer measure and is called the outer measure induced by 
.
