On the reversed sub-martingale property of empirical discrepancies in arbitrary sample spaces

The empirical discrepancy is defined as a supremum over a class of functions of a collection of centered sample averages. For uncountable classes the discrepancy need not be measurable, and distributional assertions can become dependent on the structure of the underlying probability space. This paper shows that one such assertion—the reversed sub-martingale property—is valid when interpreted in terms of measurable cover functions for the canonical model, but that it can fail in other constructions of the underlying model.