= Essentially bounded function
{title2=$|f|\leq M\text{ almost everywhere for some }M<\infty$}
= Essentially bounded
{synonym}
A <measurable function> is <essentially bounded> if some finite nonnegative <real number> $M$ bounds its <absolute value> <almost everywhere>. The exceptional <null set> may contain arbitrarily large values. With <Lebesgue measure> on $[0,1]$, a <function> equal to zero outside $\{1/n:n\in\mathbb N_{>0}\}$ and equal to $n$ at $1/n$ is <essentially bounded> despite being unbounded pointwise: its exceptional <countable set> is a <null set>, so $M=0$ works. Its <L-infinity norm> is zero because it represents the zero <equivalence class> in <Lp space>.
Contrast this with $f(x)=1/x$ on $(0,1]$. For every $M\geq1$, the values exceeding $M$ occupy $(0,1/M)$, a <set> of positive <Lebesgue measure>. Hence no finite <almost everywhere> bound works, and this <function> is not <essentially bounded>. Being finite at every input does not imply essential boundedness.
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