= Essential infimum and essential supremum
{title2=$\operatorname{ess\,inf}$, $\operatorname{ess\,sup}$}
{wiki}
The <essential infimum> and <essential supremum> describe the lower and upper bounds of a <measurable function> after disregarding <null sets> for a specified <measure>. A pointwise <infimum> or <supremum> sees every input; an essential bound sees whether values below or above a proposed threshold occur on a set of positive <measure>. Thus "essential" means insensitive to changes on a <null set>, rather than insensitive to unusual values. This distinction makes essential bounds compatible with <Lebesgue integrals> and the <equivalence classes> used in <Lp spaces>. <Measure Theory by John K. Hunter/Lp spaces (Chapter 7 of Measure Theory)>[Hunter, Measure Theory, Definitions 7.1, 7.3–7.4: almost-everywhere equivalence and essential bounds]{ref}.
For a real <measurable function> $f$ on a <measure space> $(X,\Sigma,\mu)$, the definitions are
$$
\operatorname{ess\,inf}_\mu f=\sup\{a\in\mathbb R:\mu(\{x:f(x)<a\})=0\},\qquad
\operatorname{ess\,sup}_\mu f=\inf\{b\in\mathbb R:\mu(\{x:f(x)>b\})=0\}.
$$
These can take values in the <extended real number line>. On a space of positive total <measure>, finite essential bounds obey $\operatorname{ess\,inf}_\mu f\leq f\leq\operatorname{ess\,sup}_\mu f$ <almost everywhere>. If $s=\operatorname{ess\,sup}_\mu f$ is finite, every threshold $b<s$ has $\mu(\{f>b\})>0$; the corresponding lower-edge statement is $\mu(\{f<a\})>0$ for every $a>\operatorname{ess\,inf}_\mu f$. These are the measure-based counterparts of the <approximation property of a supremum> and <approximation property of an infimum>.
Here is a two-sided example on $[0,1]$ with <Lebesgue measure>:
$$
g(x)=\begin{cases}-3,&x=1/4,\\4,&x=3/4,\\x,&\text{otherwise}.\end{cases}
$$
\b[The pointwise bounds are $\inf g=-3$ and $\sup g=4$, while the essential bounds are $\operatorname{ess\,inf}g=0$ and $\operatorname{ess\,sup}g=1$.] Both exceptional <singletons> are <null sets>. But for $0<a<1$, the inputs where $g<a$ still have <measure> $a$, and those where $g>a$ have <measure> $1-a$: deleting or adding two individual inputs cannot change those lengths. Hence neither essential bound can be moved inward.
\Image[/essential-upper-lower-bound-exceptions.png]
{title=<Essential infimum> and <essential supremum> ignore two exceptional <singletons>, while pointwise <infimum> and <supremum> retain them}
{description=The <function> follows $g(x)=x$ except for the filled values $g(1/4)=-3$ and $g(3/4)=4$. Hollow markers exclude the ordinary values at those inputs. The shaded horizontal band spans the essential bounds zero and one for <Lebesgue measure>.}
The choice of <measure> is part of the answer. The same $g$ has essential bounds $-3$ and $4$ under <counting measure>, because every nonempty <set> has positive <measure>. Under the <Dirac measure> at $1/4$, both essential bounds equal $-3$. A <null set> for one <measure> need not be a <null set> for another.
Nor must the exceptional inputs be isolated. The <Dirichlet function>, equal to one at <rational numbers> and zero at <irrational numbers>, has pointwise <supremum> one on $[0,1]$, but <essential supremum> zero for <Lebesgue measure>: the <rational numbers> form a <countable set> that is a <null set>, even though they are <dense>. Conversely, a value ten on an <interval> of length $0.001$ affects the <essential supremum> just as decisively as it would on half the domain. Small positive <measure> is not zero <measure>.
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