Assume $f_n(x)$ is a sequence of uniformly integrable functions on $[0,T]$.Is there a subsequence of $f_n(x)$ that converges to some function $f$ almost surely, i.e. $f_{n_k}(x)$ is a subsequence of $f_n(x)$ such that $f_{n_k}(x) \to f(x)$ a.e.?
Remark: the defintion of the uniform integrability is : $$\lim_{N\to\infty} \sup_{n} \int_{\{x\in[0,T]\hspace{1pt}: f_n (x) \ge N\}} f_n(x) dx = 0$$
I tried to solve it by proving it by convergence in measure, but I cannot work it out because I am not sure it has the property of convergence in $L^1$ mutually. Could you please help me ?
If there isn't such a property, could you give me a counterexample?
