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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.

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  • $\begingroup$ 1. By Egoroff's theorem we can say that on a set of finite measure, if $(f_{n_k})$ that converges point wise almost every where to $0$, then $(f_{n_k})$ converges almost uniformly to $0$. Please, note that almost uniform convergence is not the same as uniform convergence almost everywhere. $\endgroup$ – Ramiro May 7 '16 at 6:08
  • $\begingroup$ 2. It is a known result that if a sequence $(f_k)$ of lebesgue measurable functions converges to $0$ in measure then there is subsequence $(f_{n_k})$ which converges almost uniformly. As a consequence, there is a set of of finite measure where $(f_{n_k})$ converges uniformly to $0$. $\endgroup$ – Ramiro May 7 '16 at 6:25
  • $\begingroup$ @Ramiro : I understand 1st point but not second point $\endgroup$ – user311526 May 7 '16 at 7:34
  • $\begingroup$ 2. It is a known result that if a sequence $(f_k)$ of lebesgue measurable functions converges to $0$ in measure then there is subsequence $(f_{n_k})$ which converges almost uniformly to $0$. It means, for all $\varepsilon>0$, there is a set measurable set $A$ such that $m(A)<\varepsilon$ and $(f_{n_k})$ converges o $0$ uniformly on $\mathbb{R}-A$. So the answer to your question is NO, there is not a sequence as you described in your question. (See, for instance, Bartle, Elements of Integration, theorem 7.11 or Halmos, Measure Theory, section 22, theorem D). $\endgroup$ – Ramiro May 8 '16 at 22:54

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