Simplifying a bit, I encountered this notation: Where $\cal M$ and $\cal N$ are sets, each with a metric,
$[{\cal M}]^{4\epsilon} \supset [{\cal N}]^{3\epsilon}$ .
Any guesses as to the meaning of the brackets and superscript?
I'm reading $\supset$ as superset; I see no reason in the context to treat it as implication. This $\epsilon$ is used, at an earlier stage of the same theorem, to define a small distance between elements of the sets, and to define the diameter of open balls in a covering of closely related sets.
(Further background: The notation appears on p. 1885 of "A frequentist understanding of sets of measures" by Fierens, Rêgo, and Fine (pdf available here). The full expression there is: $P([\mbox{ch}({\cal M})]^{4\epsilon} \supset [\hat{\cal M}_{\Psi_U}]^{3\epsilon} \supset {\cal M}_f)\geq 1-\delta$, where "ch" means convex hull and $P$ is a probability. The contents of the $\cal M$-sets are also probability measures, as it happens. Not sure if it's worth saying more about how those three sets are defined.)