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I am having problems on how to start, any hints are appreciated. Thank you.

Show that a sequence of measurable functions $f_n$ on $\mathbb R^n$ converges to $0$ in measure if and only if $$ \lim_{n \to \infty}\left(\int \frac{|f_n|}{1+|f_n|}\right)=0. $$

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Hint: apply Markov's inequality. For any $\epsilon \in (0,1)$, we have $$ \int \frac{|f_n|}{1 + |f_n|} \geq \epsilon\cdot \mu\left\{ \frac{|f_n|}{1 + |f_n|} \geq \epsilon\right\} $$ Noting that $$ \frac{|f_n|}{1 + |f_n|} \geq \epsilon \iff |f_n| \geq \frac{\epsilon}{1-\epsilon} $$

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  • $\begingroup$ never heard of it before, is it the same as Chebyshev's inequality. thank you. $\endgroup$ – user151816 Jun 10 '14 at 15:29
  • $\begingroup$ I think so. The term Chebyshev's inequality is sometimes reserved for a more specific application. $\endgroup$ – Omnomnomnom Jun 10 '14 at 15:37

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