Let $\displaystyle (X,F, \mu)$ be a measure space with $\displaystyle \mu (X) < \infty$.

Prove that $\displaystyle f_n \rightarrow f$ in measure $\displaystyle \Longleftrightarrow$ $\displaystyle \int_X \frac{|f_n-f|}{1+|f_n-f|} d \mu$$\displaystyle \rightarrow 0$.