A measure 
on a sigma-algebra 
over 
is a complete measure if every subset of every measurable
set of measure zero is measurable. Equivalently,
if 
and 
,
then 
implies 
(and hence 
).
Every measure space has a completion, obtained by adjoining all subsets of measurable sets of measure zero. The completed sigma-algebra can be written
 |
and the measure extends uniquely by setting 
. Lebesgue
measure is complete. By contrast, Lebesgue measure restricted to the Borel
sigma-algebra is not complete, and its completion is Lebesgue
measure.
