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Is a locally bounded function $f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ also "bounded almost everywhere"?

Is the viceversa true?

Notes.

Definition of "local boundedness":

$f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ is locally bounded if for any $x \in \mathbb{R}^n$ there exists a neighborhood $A$ of $x$ such that $f(A)$ is a bounded set, that is, for some $M > 0$ we have $f(x) \leq M$ for all $x \in A$.

Definition of "almost-everywhere boundedness":

$f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ is bounded almost everywhere on $\mathbb{R}^n$ if there exist $M>0$ and a set $E \in \mathbb{R}^n$ of measure $0$ so that $f(x) \leq M$ for any $x \notin E$.

  • 6
    Did you try to write down the precise meaning of both things, put them next to each other, and stare at them for a while?2012-06-05
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    Hint: Look at $f(x) = 1/x$ on $\mathbb{R}_+$ or perhaps even better $f(x) = x$. (Then adapt to your setting.)2012-06-05
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    Good. You should be able to see that every continuous function is locally bounded. Are there continuous functions that are unbounded? Do they satisfy the second definition?2012-06-05
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    Your definition of boundedness is meaningless if range of $f$ is multidimensional2012-06-05
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    I made $f$ scalar.2012-06-05
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    So $f(x)=x$ is locally bounded but not bounded almost everywhere, right?2012-06-05
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    @Adam yes. Now, how would you modify it to get a positive valued function?2012-06-05
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    just $f(x)=|x|$2012-06-05

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