Question is :
Does there exist a sequence $\left(f_k\right)$ of Lebesgue measurable functions such that $f_k$ converges to $0$ in measure in $\mathbb{R}$ but no subsequence converges uniformly on any subset of positive measure?
See that $\left(f_k\right)$ converges to $0$ in measure implies there exists a subsequence $\left(f_{n_k}\right)$ that converges point wise almost every where to $0$.
Then by Egoroff's theorem we can say that on a set of finite measure, $\left(f_{n_k}\right)$ converges uniformly almost everywhere to $0$.
I am not sure if we can say it converges uniformly and not just almost everywhere.