= Essential supremum
{title2=$\operatorname{ess\,sup}_\mu f$}
= Essential suprema
{synonym}
The <essential supremum> is the upper bound for a <measurable function> after disregarding <null sets> for a specified <measure> $\mu$. Its definition is
$$
\operatorname{ess\,sup}_\mu f=\inf\{M\in\mathbb R:\mu(\{x:f(x)>M\})=0\},
$$
with values allowed in the <extended real number line>. A <supremum> controls every value; an <essential supremum> controls values <almost everywhere>. It therefore depends on the chosen <measure>, as well as on the <function>. https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch7.pdf[Hunter, Measure Theory, Definition 7.3: essential supremum]{ref}.
For a worked example, use <Lebesgue measure> on $[0,1]$ and put
$$
f(x)=\begin{cases}10,&x=1/2,\\x,&x\ne1/2.\end{cases}
$$
Its <supremum> is ten, attained at $1/2$. But \b[its <essential supremum> is one]: $f\leq1$ outside the <singleton> $\{1/2\}$, which is a <null set>. Every candidate $M<1$ fails, because the <interval> $(\max(0,M),1)$ has positive <Lebesgue measure> and includes values larger than $M$; changing one point cannot remove that interval. In particular, $\mu(\{f>0.8\})=0.2$, while $\mu(\{f>1\})=0$.
\Image[/essential-bounds-ignore-null-sets.png]
{title=A <supremum> sees an isolated exceptional value; an <essential supremum> depends on positive <measure>}
{description=Left: $f(1/2)=10$, with the ordinary value $1/2$ excluded at that input. Right: a zoom shows that values above $0.8$ still occur on an <interval> of <Lebesgue measure> $0.2$. The isolated value at ten lies outside the zoom. Thus the <essential supremum> is one, despite the pointwise <supremum> being ten.}
Why is this useful? <Lebesgue integrals> ignore changes on <null sets>, and <Lp spaces> identify <functions> that agree <almost everywhere>. The <L-infinity norm> uses $\operatorname{ess\,sup}|f|$ so it respects that identification. For this nonnegative example the <supremum norm> is ten and the <L-infinity norm> is one; replacing $f(1/2)$ by $1/2$ changes the former while leaving the latter unchanged. https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch7.pdf[Hunter, Measure Theory, Definition 7.4: the L-infinity norm]{ref}.
The <measure> qualification is essential. For $\mu=\delta_{1/2}$, the <Dirac measure> concentrated at the exceptional input, that <singleton> has <measure> one and this <function> has <essential supremum> ten. A set ignored by one <measure> need not be ignored by another. The lower counterpart is the <essential infimum>, with $\operatorname{ess\,inf}f=-\operatorname{ess\,sup}(-f)$.
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