##  [Essential Supremum](/essential-supremum) 

  ##  [Essential Supremum](https://puremath.quantumdictionary.io/essential-supremum-0) 

  

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- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

For a measurable function f on a measure space, the essential supremum is the smallest extended real number M such that f(x) ≤ M for almost every x (i.e., except on a set of measure zero). It ignores exceptional values on null sets and therefore yields an a.e. upper bound rather than a pointwise maximum.