An outer measure on a set 
is a set function ![mu^*:2^X->[0,infty]](/images/equations/OuterMeasure/Inline2.svg)
such that 
and that 
is countably monotone,
where 
is the empty set.
Given a set 
of subsets of 
and an arbitrary set function ![mu:S->[0,infty]](/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 finite or countable families 
of sets in 
that cover 
, meaning 
. If there is no such family, the infimum
is taken to be 
.
The resulting function ![mu^*:2^X->[0,infty]](/images/equations/OuterMeasure/Inline17.svg)
is an outer measure and is called the
outer measure induced by 
.
