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.