The essential supremum is the proper generalization to measurable functions of the maximum. The technical difference is that the values of a function on a set of measure zero don't affect the essential supremum.
Given a measurable function 
, where 
is a measure space with measure
,
the essential supremum is the smallest number 
such that the set
 |
has measure zero. If no such number exists, as in the case of 
on 
,
then the essential supremum is 
.
The essential supremum of the absolute value of a function 
is usually denoted 
, and this serves as the norm for L-infty-space.
