
Convergence in measure
Let $\displaystyle X \subseteq \mathbb{R}$ and $\displaystyle m$ to be defined as Lebesgue measure.Suppose for each $\displaystyle \alpha >0$ , we have $\displaystyle \lim m(\{x \in X : f_n(x)f(x) \le \alpha \} =0$. Can we conclude from here that the sequence $\displaystyle (f_n)$ converges in measure to $\displaystyle f$?
Here $\displaystyle (f_n)$ convergence in measure to $\displaystyle f$ is defined as for each $\displaystyle \alpha >0$ ,we have $\displaystyle \lim m(\{x \in X : f_n(x)f(x) \ge \alpha \} =0$.
My guess is no. But I am unable to construct a counterexample. Can anyone help?