Essential Supremum
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.
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