3.3. Uniform Law Under Random Entropy

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Using all the lemmas from 3.2 to 3.5, we prove another sufficiency of the uniform law: the vanishing random entropy condition. This condition is fairly weaker than the finite bracketing entropy condition in two senses. First, instead of the (sup-normed) envelope condition, it only requires the envelope to be integrable. In $L^p(Q)$ where $Q$ is a finite measure, this is clearly implied by the envelop condition. In addition, as we already saw before, the vanishing random entropy is implied by the finite bracketing entropy condition.
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