# When does Almost sure convergence imply Almost sure uniform convergence?

Suppose that on a probability space $$(\Omega,\mathcal{F},\mathbb{P})$$, a sequence of random variables $${X}_n$$ converges to $$0$$ almost surely. Fix $$\epsilon >0$$. By definition of almost sure convergence, we know that there exists a set $$\Omega_0 \subseteq \Omega$$, such that (i) $$\mathbb{P}(\Omega_0)=1$$, and (ii) for each $$\omega\in\Omega_0$$, there exists $$N(\omega,\epsilon)$$ s.t. $$\forall n \geq N(\omega,\epsilon), |X_n(\omega)| \leq \epsilon.$$ My question is when can we conclude anything about the uniformity of $$N(\omega,\epsilon)$$, i.e., under what conditions can we obtain an $$N(\epsilon)$$ that is $$\omega$$-independent?

Egorov's theorem tells us that such an assertion regarding uniform convergence can be made for a set that has measure arbitrarily close to 1. I was wondering if there are additional conditions under which we can claim such an assertion for a set of measure 1.