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,