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Fix a function $\phi \in C_c(\mathbb{R}), \phi\not\equiv0,$ and consider the family of functions

$\mathcal{F} = \{\phi_n:n\in\mathbb N\},$ where $\phi_n(x) = \phi(x+n), x\in\mathbb{R}.$

The problem is to prove that $\mathcal{F}$ does not have compact closure in $L^p(\mathbb{R}) (1\le p <\infty),$ but there is a theorem that the closure $\mathcal{F}|_{\Omega}$ in $L^p(\Omega)$ is compact for any finite-measure subset $\Omega\in\mathbb{R}.$ Thus this problem is intended to show that the theorem is not applicable to infinite-measure sets, in particular, $\mathbb{R}.$

I think I need to induce contradiction assuming $\mathcal{F}$ has compact closure in $L^p(\mathbb{R}),$ but I cannot come up with the next step.

Any hint or idea would be appreciated. Thanks in advance.

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Hint: We have $\operatorname{supp}\phi\subset [-N,N]$ for some $N\in\mathbb N$. Can the sequence $\{\phi_{nN}\}_{n\in\mathbb N}$ have a convergent subsequence?

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  • $\begingroup$ Your answer makes me feel I am stupid... Thank you Aweygan! $\endgroup$ – Euduardo Nov 25 '18 at 5:33
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    $\begingroup$ You're welcome, but don't feel stupid! $\endgroup$ – Aweygan Nov 25 '18 at 5:36

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