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Let $f: \mathbb{R}\to\mathbb{R}$ and the essential supremum of $f$ $$e=\inf\{\alpha\ge 0 \mid |f|\le\alpha\text{ almost everywhere}\}$$

I can't see why $$\lambda^*\left(\{x\in\mathbb{R} \mid |f(x)|\ge e-\epsilon\}\right)>0$$ for $a0$, where $\lambda$ is the Lebesgue measure.

Could someone help me ?

Fixed after Didier Piau's comment.

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    The assertion is false. It is true if you replace $(a,b)$ by $\mathbb{R}$.2011-03-27
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    Or the statement should be: *There are* $a$ and $b$ such that $\ldots >0$.2011-03-27
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    @Didier Piau: Thank you, I fixed the question.2011-03-27

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if $\alphanot less or equal $\alpha$ a.e. (else $\alpha\geq e$ by definition), so there is a set of positive measure where $|f|>\alpha$.

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    Thank you for the answer and sorry for the question, I should have taken a pause...2011-03-27