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Let $f:\Omega \rightarrow \mathbb{R} $ be a function in $L^p(\Omega)$ for some $1\leq p < \infty$ and $g: \mathbb{R} \rightarrow[a,b]$ continuous, with $a,b\in \mathbb{R}$.

Do we have $g \circ f \in L^\infty(\Omega)$? Since $\sup\limits_{\mathbb{R}}g <\infty $ we should have $\| g \circ f \|_{L^\infty(\Omega)} < \infty$ so the answer should be yes, but I have the feeling I'm missing something here.

Thank you in advance for any hint.

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Of course. For all $t\in\mathbb{R}$, $g(t)\in [a,b]$, so for all $x\in\Omega$, $g(f(x))\in[a,b]$. So $g\circ f$ is indeed a bounded function.

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  • $\begingroup$ Thank you very much. This sounded too good to be true, so I was not sure! $\endgroup$ – jason paper Sep 6 '18 at 20:26
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The slightly less trivial fact is that $g \circ f$ is measurable. Note that this would not necessarily be true if you only assumed $g: \mathbb R \to [a,b]$ was (Lebesgue) measurable: e.g. see this post.

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  • $\begingroup$ Thank you, useful hint! $\endgroup$ – jason paper Sep 6 '18 at 20:42

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