<s>[INST] Sequence of continuous, integrable functions that are not dominated I am looking for a sequence of continuous functions ${f_n}: [0,1] \rightarrow [0,\infty]$ such that $\int f_n d\mu \rightarrow 0$, $f_n(x) \rightarrow 0$ for all $x$, but $f(x) = \sup f_n(x)$ is not in $L_1$.
The "rotating tower" functions with growing heights seem to not converge to zero. A function that rises over reducing intervals, such as $f_n(x) = \sqrt{n} {\bf 1}_{[0,\frac{1}{n})}$ wouldn't satisfy the last condition…
Would you have any suggestions on how I could find such functions?
 [/INST] Let
$$f_n(x) = \begin{cases}\frac{1}{x} &, \frac{1}{n+1} \leqslant x < \frac{1}{n}\\
(n+1)2^{n+1}\left(x -\frac{1}{n+1} + \frac{1}{2^{n+1}}\right) &, \frac{1}{n+1} - \frac{1}{2^{n+1}} \leqslant x < \frac{1}{n+1}\\
n2^n\left(\frac1n + \frac{1}{2^n} - x\right) &, \frac{1}{n} < x < \frac1n + \frac{1}{2^n}\\
0 &, \text{ otherwise}. \end{cases}$$
Then $f_n \to 0$ pointwise, $\int_0^1 f_n(x)\,dx = \log (1 + 1/n) + \frac{n}{2^{n+1}} + \frac{n+1}{2^{n+2}} \to 0$, but $\sup_n f_n(x) = \frac1x \notin L^1$.
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