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Suppose $f_1(x),f_2(x),\ldots:[0,1]\rightarrow\mathbb{R}$ are measurable functions such that $f_1(x)\geq f_2(x)\geq\ldots$. (infinite sequence) and $\lim_{n\rightarrow\infty}f_n(x)=0$. Is it true that $\lim_{n\rightarrow \infty}\int_0^1 f_n(x)dx=0$?

I wanted to apply the dominated convergence theorem but cannot, because $f_i(x)$ is not necessarily integrable. So is there a counterexample for this?

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$$ f_n(x) = \begin{cases} \dfrac{\chi_{[0, 1/n]}}{x} &: x \in (0, 1] \\ 0 &: x = 0\end{cases} $$

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  • $\begingroup$ Is $\chi_[0,1/n]$ the characteristic function that takes the value $1$ on $[0,1/n]$ and $0$ otherwise? $\endgroup$ – PJ Miller Oct 28 '13 at 23:41
  • $\begingroup$ @PJMiller Yes! ${}$ $\endgroup$ – Ayman Hourieh Oct 28 '13 at 23:41

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