= Essential range
{title2=$\operatorname{essran}_\mu m$}
{wiki}
The essential range of a real or complex <measurable function> $m$ consists of the values $z$ whose every <neighborhood> receives inputs of positive <measure>:
$$
\operatorname{essran}_\mu m=\{z:\mu(\{x:|m(x)-z|<\varepsilon\})>0\text{ for every }\varepsilon>0\}.
$$
It depends on the <measure> as well as the <function>. This definition concerns values arbitrarily near $z$, not necessarily inputs at which $m=z$ exactly. It therefore retains meaningful limiting values while ignoring values occurring only on <null sets>. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe2.pdf[Teschl, Mathematical Methods in Quantum Mechanics, §2.4, equation (2.77): essential range]{ref}.
For $m(x)=x$ on $(0,1)$ with <Lebesgue measure>, \b[the ordinary <image of a function> is $(0,1)$, but the <essential range> is $[0,1]$]. Neither endpoint is attained, yet inputs with $|m(x)|<\varepsilon$ or $|m(x)-1|<\varepsilon$ have positive <measure> for every $\varepsilon>0$. Every interior value also belongs to the <essential range>, although its exact <fiber> is a <singleton> of zero <measure>. Positive <measure> of a fiber is sufficient, but is not necessary, for a value to belong to the <essential range>.
The two-sided example $g$ from <essential infimum and essential supremum> illustrates what disappears. It follows $g(x)=x$ on $[0,1]$, except $g(1/4)=-3$ and $g(3/4)=4$. Its ordinary <image of a function> is $([0,1]\setminus\{1/4,3/4\})\cup\{-3,4\}$, whereas its <essential range> remains $[0,1]$. The missing ordinary values $1/4$ and $3/4$ still have arbitrarily close neighbors with positive <measure>; the exceptional values $-3$ and $4$ can each be isolated by a sufficiently small <neighborhood> whose <preimage> is a <null set>.
\Image[/essential-upper-lower-bound-exceptions.png]
{title=The <essential range> of the modified identity <function> remains $[0,1]$, while its ordinary <image of a function> includes two isolated outliers}
{description=For <Lebesgue measure>, the filled outliers $-3$ and $4$ fail the criterion of positive <measure> in every <neighborhood>. The hollow values $1/4$ and $3/4$ still belong to the <essential range> because nearby outputs occur on <intervals> of positive <measure>.}
For the three-point example in <L-infinity norm>, the values are $1,3,10$. With weights $(0.5,0.5,0)$ the <essential range> is $\{1,3\}$; with weights $(0.499,0.5,0.001)$ it is $\{1,3,10\}$. A small but positive <probability> retains ten, so an <essential range> does not discard a value merely because it is rare. For a real <function> on a space of positive total <measure>, the <essential infimum> and <essential supremum> are the lower and upper edges of its <essential range>, allowing infinite bounds. The <L-infinity norm> instead takes the upper edge of the <essential range> of $|m|$.
\Image[/lp-norm-null-versus-rare-value.png]
{title=Including ten in the <essential range> changes the limiting <Lp norm>, even when its <probability> is only $0.001$}
{description=The blue case has <essential range> $\{1,3\}$, while the red case has <essential range> $\{1,3,10\}$. These finite positive-valued examples make their largest essential values equal to their <L-infinity norms>.}
A useful operator connection is the <spectrum of a real multiplication operator>: on a nonzero <L2 space> over a <sigma-finite measure>, multiplication by a real <measurable function> $m$ has <spectrum> equal to its <essential range>, with the natural domain when $m$ is unbounded. Outside that range, $|m-z|$ has a positive lower bound <almost everywhere>, so multiplication by $1/(m-z)$ gives an <inverse function> that is a <bounded linear operator>. Near an essential value, this lower bound fails. Thus discarding <null sets> is precisely what makes the range match an operator acting on <almost everywhere> <equivalence classes>. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe2.pdf[Teschl, Mathematical Methods in Quantum Mechanics, §2.4, equations (2.74)–(2.77): multiplication-operator spectrum]{ref}.
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