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Consider a sequence of continuous integrable functions $\{f_n(t)\}_n$ such that
$\ast\ \displaystyle\lim_{n\rightarrow\infty}f_n(t)=0$ for all $t>0$
$\ast\ \{f_n(t)\}_n$ is such that $f_n(t)<f(t)$ for all $n$ and $t>0$ where $f\in L^1([0,\infty[)$
$\ast\ \displaystyle\lim_{n\rightarrow\infty}n^2f_n(t)=g(t)$ for all $t>0$ where $g\in L^1([0,\infty[)$
Then is it true that there exists a function $h\in L^1([0,\infty[)$ such that $n^2f_n(t)<h(t)$ for all $n$ and $t>0$?
Or can it be true if we add some hypothesis on $f_n$?

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Consider the sequence of functions

$$ f_n = 1/n^2 \cdot \chi_{(n,n+1)}, $$

where $\chi_A$ is the indicator function of the set $A$.

I will leave the details for you.

The important observation is that the existence of a function $h$ as in your hypothesis would allow you to apply dominated convergence. This should help you to show that there is no such function. (One can also verify it directly).

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  • $\begingroup$ Is your example bounded by an integrable function? $\endgroup$ – anonymous Oct 30 '14 at 12:53
  • $\begingroup$ Yes, by the sum of all $f_n$. $\endgroup$ – PhoemueX Oct 30 '14 at 13:32

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