  Fractal-like structures of varying complexity are common in nature, and
measure-based dimensions (Minkowski, Hausdorff) supply their basic geometric
characterization. However, at the level of fundamental dynamics, which is
quantum, structure does not enter via geometry of fixed sets but is encoded in
probability distributions on associated spaces. The question then arises
whether a robust notion of fractal measure-based dimension exists for
structures represented in this way. Starting from effective number theory, we
construct all counting-based schemes to select effective supports on
collections of objects with probabilities and associate the effective counting
dimension (ECD) with each. We then show that ECD is scheme-independent and,
thus, a well-defined measure-based dimension with meaning analogous to the
Minkowski dimension of fixed sets. In physics language, ECD characterizes
probabilistic descriptions arising in a theory or model via discrete
``regularization''. For example, our analysis makes recent surprising results
on effective spatial dimensions in quantum chromodynamics and Anderson models
well founded. We discuss how to assess the reliability of regularization
removals in practice and perform such analysis in the context of 3d Anderson
criticality.
