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.

My guess is no. But I am unable to construct a counterexample. Can anyone help?