Examples
On the real line consider the Lebesgue measure and its corresponding σ-algebra Σ. Define a function f by the formula
The supremum of this function (largest value) is 5, and the infimum (smallest value) is −4. However, the function takes these values only on the sets {1} and {−1} respectively, which are of measure zero. Everywhere else, the function takes the value 2. Thus, the essential supremum and the essential infimum of this function are both 2.
As another example, consider the function
where Q denotes the rational numbers. This function is unbounded both from above and from below, so its supremum and infimum are ∞ and −∞ respectively. However, from the point of view of the Lebesgue measure, the set of rational numbers is of measure zero; thus, what really matters is what happens in the complement of this set, where the function is given as arctan x. It follows that the essential supremum is π/2 while the essential infimum is −π/2.
Lastly, consider the function f(x) = x3 defined for all real x. Its essential supremum is +∞, and its essential infimum is −∞.
Read more about this topic: Essential Supremum And Essential Infimum
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