Essential infimum and essential supremum

While the exact definition is not immediately straightforward, intuitively the essential supremum of a function is the smallest value that is greater than or equal to the function values everywhere while ignoring what the function does at a set of points of measure zero. For example, if one takes the function ${\displaystyle f(x)}$ that is equal to zero everywhere except at ${\displaystyle x=0}$ where ${\displaystyle f(0)=1,}$ then the supremum of the function equals one. However, its essential supremum is zero since (under the Lebesgue measure) one can ignore what the function does at the single point where ${\displaystyle f}$ is peculiar. The essential infimum is defined in a similar way.

Definition

As is often the case in measure-theoretic questions, the definition of essential supremum and infimum does not start by asking what a function ${\displaystyle f}$ does at points ${\displaystyle x}$ (that is, the image of ${\displaystyle f}$), but rather by asking for the set of points ${\displaystyle x}$ where ${\displaystyle f}$ equals a specific value ${\displaystyle y}$ (that is, the preimage of ${\displaystyle y}$ under ${\displaystyle f}$).

Let ${\displaystyle f:X\to \mathbb{R}}$ be a real valued function defined on a set ${\displaystyle X.}$ The supremum of a function ${\displaystyle f}$ is characterized by the following property: ${\displaystyle f(x)\le supf\le \mathrm{\infty }}$ for all ${\displaystyle x\in X}$ and if for some ${\displaystyle a\in \mathbb{R}\cup \{+\mathrm{\infty }\}}$ we have ${\displaystyle f(x)\le a}$ for all ${\displaystyle x\in X}$ then ${\displaystyle supf\le a.}$ More concretely, a real number ${\displaystyle a}$ is called an upper bound for ${\displaystyle f}$ if ${\displaystyle f(x)\le a}$ for all ${\displaystyle x\in X;}$ that is, if the set $${\displaystyle f^{-1}(a,\mathrm{\infty })=\{x\in X:f(x)>a\}}$$ is empty. Let $${\displaystyle U_f=\{a\in \mathbb{R}:f^{-1}(a,\mathrm{\infty })=\varnothing \}}$$ be the set of upper bounds of ${\displaystyle f}$ and define the infimum of the empty set by ${\displaystyle inf\varnothing =+\mathrm{\infty }.}$ Then the supremum of ${\displaystyle f}$ is $${\displaystyle supf=inf{U}_f.}$$ By definition, if the set of upper bounds ${\displaystyle U_f}$ is empty, we have ${\displaystyle supf=+\mathrm{\infty }}$.

Now assume in addition that ${\displaystyle (X,\mathrm{\Sigma },\mu )}$ is a measure space and, for simplicity, assume that the function ${\displaystyle f}$ is measurable. Similar to the supremum, the essential supremum of a function is characterised by the following property: ${\displaystyle f(x)\le \mathrm{ess}supf\le \mathrm{\infty }}$ for ${\displaystyle \mu }$-almost all ${\displaystyle x\in X}$ and if for some ${\displaystyle a\in \mathbb{R}\cup \{+\mathrm{\infty }\}}$ we have ${\displaystyle f(x)\le a}$ for ${\displaystyle \mu }$-almost all ${\displaystyle x\in X}$ then ${\displaystyle \mathrm{ess}supf\le a.}$ More concretely, a number ${\displaystyle a}$ is called an essential upper bound of ${\displaystyle f}$ if the measurable set ${\displaystyle f^{-1}(a,\mathrm{\infty })}$ is a set of ${\displaystyle \mu }$-measure zero, That is, if ${\displaystyle f(x)\le a}$ for ${\displaystyle \mu }$-almost all ${\displaystyle x}$ in ${\displaystyle X.}$ Let $${\displaystyle U_f^{\mathrm{ess}}=\{a\in \mathbb{R}:\mu (f^{-1}(a,\mathrm{\infty }))=0\}}$$ be the set of essential upper bounds. Then the essential supremum is defined similarly as $${\displaystyle \mathrm{ess}supf=inf{U}_f^{\mathrm{e}\mathrm{s}\mathrm{s}}}$$ if ${\displaystyle U_f^{\mathrm{ess}}\ne \varnothing ,}$ and ${\displaystyle \mathrm{ess}supf=+\mathrm{\infty }}$ otherwise.

Exactly in the same way one defines the essential infimum as the supremum of the essential lower bounds, that is, $${\displaystyle \mathrm{ess}inff=sup\{b\in \mathbb{R}:\mu (\{x:f(x)<b\})=0\}}$$ if the set of essential lower bounds is nonempty, and as ${\displaystyle -\mathrm{\infty }}$ otherwise; again there is an alternative expression as ${\displaystyle \mathrm{ess}inff=sup\{a\in \mathbb{R}:f(x)\ge a\text{ for almost all }x\in X\}}$ (with this being ${\displaystyle -\mathrm{\infty }}$ if the set is empty).