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I feel a bit puzzled about the difference between these two definitions:

  • $f$ is uniform bounded if $\vert f(x) \vert \le M\,\forall x$ for some constant $M \gt 0$ independent of $x$.
  • $f$ is essential bounded if ${\vert \vert f \vert \vert}_{L^\infty} \equiv\inf\{ C\gt 0: \vert f(x) \vert \le c \;for\;a.e\;x\} \lt \infty$

Is it true that the first one implies the second one but not vice versa? If the answer is positive then,the reason is that uniform boundedness implies pointwise boundedness or not? How can I distinguish these two definitions?

Any help is appreciated! Thanks in advance

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  • $\begingroup$ What is definition of f is uniformly bounded? And what is f is bounded?. Is there any difference? $\endgroup$ – Avinash N Oct 29 '18 at 13:07
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Clearly, the first implies the second one. The converse is not true. Consider $$ f(x) = \begin{cases} x & \text{ if } x\in \mathbb Z,\\ 0 & \text{ else}. \end{cases}. $$ Then $f$ is essentially bounded ($f(x)=0$ for almost all $x$, but not uniformly bounded.

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