I have been looking into Martin-Löf random (MLR) sequences. The intuition is that when a statement $P$ is true for almost all sequences, it holds for ALL MLR sequences. For example, the strong law of large numbers holds for all MLR random sequences, meaning that if $\omega$ is a binary MLR sequence (according to the standard measure on the Cantor space $2^{\mathbb{N}})$, then $$ \lim_{n\rightarrow \infty}\frac{ \# \{ 1'\text{s in the first } n \text{ digits of } \omega \} }{n}=\frac{1}{2}. $$ Written differently, if $f_n(\omega)=\frac{1}{n}\sum_{i\leq n}\omega_i -1/2$, then $\lim_{n\rightarrow \infty} f_n(\omega)\rightarrow 0$ for all MLR sequences.
My question is, does this idea hold for every sequence of functions $f_n$? Let $(f_n)$ be a sequence of functions such that $(n, \omega)\mapsto f_n(\omega)$ is computable (in the intuitive sense). Also assume that for almost all $\omega\in 2^{\mathbb{N}}$, $\lim_{n\rightarrow \infty} f_n(\omega)\rightarrow 0$.
Is there a known counterexample to $\lim_{n\rightarrow \infty} f_n(\omega)\rightarrow 0$ for all MLR sequences $\omega$?