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I am wondering if one can find a sequence of functions $f_{n}(x)$ such that uniformly in $n$, we have $f_n (x) \in L^1(\mathbb R)$ and $\lim_{n \rightarrow \infty} f_n (x)=1$ and $ \lim_{n \rightarrow \infty}\Vert f_n \Vert_{L^{1}}=1$ or equal to another finite constante $C>0$.

Thanks.

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  • $\begingroup$ @John $\mathbb R$ $\endgroup$ – Lee-Jin Jun 8 '15 at 2:22
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    $\begingroup$ Do you mean $n \to \infty$ instead of $n \to 0$? $\endgroup$ – Robert Israel Jun 8 '15 at 2:23
  • $\begingroup$ @RobertIsrael yes !! sorry $\endgroup$ – Lee-Jin Jun 8 '15 at 2:27
  • $\begingroup$ If it's going pointwise or a.s. to 1 over all of $\mathbb{R}$, why would you think the limit of the norms would be finite? $\endgroup$ – Batman Jun 8 '15 at 2:34
  • $\begingroup$ @Batman Just a guess with a Gaussian function, probably the limit could be infinite to $\endgroup$ – Lee-Jin Jun 8 '15 at 2:38
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By Fatou's lemma, if $\lim_{n \to \infty} |f_n(x)| = 1$ almost everywhere then $$ \lim \inf_{n \to \infty} \|f_n \|_1 \ge \int_{\mathbb R} 1\; dx = \infty $$

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  • $\begingroup$ Thanks for your nice answer ! $\endgroup$ – Lee-Jin Jun 8 '15 at 2:40

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