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 f:X->R, where X is a measure space with measure mu, the essential supremum is the smallest number alpha such that the set
| {x:f(x)>alpha} |
has measure zero. If no such number exists, as in the case of f(x)=1/x on (0,1), then the essential supremum is infty.
The essential supremum of the absolute value of a function |f| is usually denoted ||f||_infty, and this serves as the norm for L-infty-space.
See also
L-infty-Space, L-p-Space, L2-Space, Measure, Measurable Function, Measure Space, SupremumThis entry contributed by Todd Rowland
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Rowland, Todd. "Essential Supremum." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/EssentialSupremum.html