We define a notion of entropy for an infinite family $\mathcal{ C}$ of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for $\mathcal{ C}$ under every ergodic transformation if and only if $\mathcal{ C}$ has zero entropy. When the entropy of $\mathcal{ C}$ is positive, we establish a strong converse showing that the uniform mean ergodic theorem fails generically in every isomorphism class, including the isomorphism classes of Bernoulli transformations. As a corollary of these results, we establish that every strong mixing transformation is uniformly strong mixing on $\mathcal{ C}$ if and only if the entropy of $\mathcal{ C}$ is zero, and obtain a corresponding result for weak mixing transformations.
Categories: math.DS