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essential supremum (Definition)

Let $ (X, \mathfrak{B}, \mu)$ be a measure space and let $ f:X \to \mathbb{R}$ be a function. The essential supremum of $ f$ is the smallest number $ a \in \mathbb{R}$ for which $ f$ only exceeds $ a$ on a set of measure zero. This allows us to generalize the maximum of a function in a useful way.

More formally, we define $ \mathrm{ess } \sup f$ as follows. Let $ a \in \mathbb{R}$, and define

$\displaystyle M_{a} = \{x: f(x) > a\},$ (1)

the subset of $ X$ where $ f(x)$ is greater than $ a$. Then let
$\displaystyle A_{0} = \{a \in \mathbb{R}: \mu(M_a) = 0\},$ (2)

the set of real numbers for which $ M_a$ has measure zero. If $ A_0 = \emptyset$, then the essential supremum is defined to be $ \infty$. Otherwise, the essential supremum of $ f$ is
$\displaystyle \mathrm{ess } \sup f :=\inf A_0.$ (3)



"essential supremum" is owned by drummond.
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See Also: supremum, $L^p$-space

Other names:  ess-sup, ess sup
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Cross-references: real numbers, subset, measure zero, number, function, measure space
There are 4 references to this entry.

This is version 1 of essential supremum, born on 2002-02-17.
Object id is 2044, canonical name is EssentialSupremum.
Accessed 12978 times total.

Classification:
AMS MSC: 28C20 (Measure and integration :: Set functions and measures on spaces with additional structure :: Set functions and measures and integrals in infinite-dimensional spaces )
Pending Errata and Addenda
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continuous f by jujutsuka on 2005-10-24 13:07:09
why does this reduce to the usual notion of supremum when f is continuous?
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