Given a measure space $\Omega$, a sequence of measurable functions $(f_n : \Omega \rightarrow \mathbb{R})_{n=1}^\infty$ and $r \in \mathbb{R}$, such that:
$\{ \omega \in \Omega : \limsup_{n \rightarrow \infty} f_n(\omega) > r \}$
has positive measure. Is it possible to find a subsequence of arbitrarily fast growing indices, such that this set still has positive measure?
To be more precise, imagine also being given a sequence $(N_k)_{k=1}^\infty$ of natural numbers. Does there always exist a sequence of natural numbers $(n_k)_{k=1}^\infty$ satisfying:
- $n_k \geq N_k$, for all $k \in \mathbb{N}^{\geq 1}$
- $\{ \omega \in \Omega :\lim_{K \rightarrow \infty} (\sup \{f_{n_K}, f_{n_{K+1}}, f_{n_{K+2}}, …\}(\omega))> r \}$
has positive measure?