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existence of the essential supremum
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(Theorem)
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We state the existence of the essential supremum for a set $\mathcal{S}$ of extended real valued functions on a $\sigma$ finite measure space $(\Omega,\mathcal{F},\mu)$
Theorem Suppose that the measure space $(\Omega,\mathcal{F},\mu)$ is $\sigma$ finite. Then, the essential supremum of $\mathcal{S}$ exists. Furthermore, if $\mathcal{S}$ is nonempty then there exists a sequence $(f_n)_{n=1,2,\ldots}$ in $\mathcal{S}$ such that \begin{equation}\label{eq:1} \operatorname{esssup}\mathcal{S}=\sup_n f_n. \end{equation}
Note that, by reversing the inequalities, this result also applies to the essential infimum, except that equation (![[*]](http://images.planetmath.org:8080/cache/objects/11399/js//usr/share/latex2html/icons/crossref.png "[*]") ) is replaced by \begin{equation*} \operatorname{essinf}\mathcal{S}=\inf_nf_n. \end{equation*}
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"existence of the essential supremum" is owned by gel.
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Cross-references: equation, essential infimum, inequalities, sequence, essential supremum, measure space, functions, real
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This is version 3 of existence of the essential supremum, born on 2008-12-27, modified 2009-02-01.
Object id is 11399, canonical name is ExistenceOfTheEssentialSupremum.
Accessed 547 times total.
Classification:
| AMS MSC: | 28A20 (Measure and integration :: Classical measure theory :: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence) |
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Pending Errata and Addenda
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