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I have difficulty in understanding the difference of the concepts "finite almost everywhere" and having finite essential supremum (as in $\Vert f \rVert_\infty= inf\{a \geq 0: \mu(\{x : |f(x)| >a \}) =0\} < \infty$.

I know that if a function f is integrable then it is finite almost everywhere. Is $\lVert f \rVert_\infty < \infty $ the same as being bounded almost everywhere?

A function such as $\frac{1}{\sqrt{x}}$ is finite almost everywhere on $[0,1]$ (since it is integrable) but not bounded almost everywhere?

I would be very greatful if some one could help me and explain the difference and maybe come with some examples.

Kind regards,

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    $\begingroup$ The $|x|^{-\frac12}$ example you gave is the key. Note that integrability is not needed. $|x|^{-50}$ is another perfectly good example of a function that is finite almost everywhere and that is not essentially bounded. $\endgroup$ Mar 15, 2017 at 9:32
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    $\begingroup$ "Finite almost everywhere" means for almost all $x$, there exists a bound $M_x$ such that $|f(x)| \leq M_x$. "Having finite ess sup" means that there exists a bound $M$ such that for almost all $x$, $|f(x)| \leq M$. $\endgroup$
    – Kenny Wong
    Mar 15, 2017 at 9:47

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