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Does there exist a decreasing sequence of Lebesgue-measurable non-negative functions $(f_n)_{n}$ such that $f_n \to 0$ pointwise on $\mathbb{R}$, but $f_n$ does not converge to $0$ in mean, by which I mean $$ \int_\mathbb{R} f_n d\mathcal{L}^1 \not\to 0 \text{ as } n \to \infty ?$$

I tried finding a counterexample, but I did not succeed. There is a theorem that says that if $|f_n| \leq g$ for every $n$ and $\displaystyle \int_\mathbb{R} |g| d\mathcal{L}^1 < \infty,$ then $\displaystyle \int_\mathbb{R} f_nd\mathcal{L}^1 \to 0$ as $n \to \infty$, but here we don't have the conditions that the sequence of functions is dominated by an integrable function.

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    $\begingroup$ Consider $\frac f n$ where $f$ is any nonnegative measurable function which is not integrable. $\endgroup$ Oct 3 '19 at 9:19
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Take $f_n=1_{[n,+\infty)}$

$f_n$ is deacreasing also

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  • $\begingroup$ Thank you! I haven't thought of this. $\endgroup$
    – S.T.
    Oct 3 '19 at 9:18

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