Convergence in measure

Let $X \subseteq \mathbb{R}$ and $m$ to be defined as Lebesgue measure.Suppose for each $\alpha >0$ , we have $\lim m(\{x \in X : |f_n(x)-f(x)| \le \alpha \} =0$. Can we conclude from here that the sequence $(f_n)$ converges in measure to $f$?
Here $(f_n)$ convergence in measure to $f$ is defined as for each $\alpha >0$ ,we have $\lim m(\{x \in X : |f_n(x)-f(x)| \ge \alpha \} =0$.