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.


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