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 infimum,
in the affinely extended real numbers,
of all real numbers 
such that the set
 |
has measure zero. If there are no such numbers, as for 
on 
with Lebesgue measure, the essential supremum
is 
.
If every real number qualifies, as on a space of measure zero, the essential supremum is 
.
When 
,
the essential supremum of the absolute value 
of a function
gives the norm of 
when 
belongs to L∞-space.
This is denoted 
.
For any measurable function, the quantity
is defined as the infimum of the nonnegative real bounds 
for which 
has measure
zero, with value 
if there are no such bounds. This convention gives 
on a space of measure
zero.
