Uniform Approximation of Vapnik-Chervonenkis Classes

For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik-Chervonenkis (VC) dimension or (ii) for every epsilon > 0 there is a finite partition pi such the pi-boundary of each set has measure at most epsilon. Immediate corollaries include the fact that a family with finite VC dimension has finite bracketing numbers, and satisfies uniform laws of large numbers for every ergodic process. From these corollaries, we derive analogous results for VC major and VC graph families of functions.

💡 Research Summary

The paper establishes a sharp dichotomy for any family 𝔽 of measurable subsets of a probability space (𝔛, 𝔅, μ). Either (i) 𝔽 has infinite Vapnik‑Chervonenkis (VC) dimension, or (ii) for every ε > 0 there exists a finite measurable partition π of 𝔛 such that the π‑boundary of each set A ∈ 𝔽 has μ‑measure at most ε. The π‑boundary ∂πA consists of those partition cells that intersect both A and its complement; thus a small boundary means the partition almost separates the set from its complement.

The main theorem (Uniform Approximation Theorem) is proved by exploiting the combinatorial properties of VC classes. If the VC dimension is infinite, Sauer‑Shelah’s lemma guarantees that for any finite partition there is a set whose boundary retains a non‑negligible measure, precluding uniform approximation. Conversely, when the VC dimension d is finite, the growth function of 𝔽 is polynomial in n, which allows the construction of an ε‑net for the class of boundaries. By refining a coarse partition into O(ε⁻¹) cells and then further subdividing according to the finite shattering capacity, one obtains a partition π with μ(∂πA) ≤ ε for all A ∈ 𝔽. The proof uses standard tools such as Vitali covering, Carathéodory extension, and the construction of ε‑brackets from the partition cells.

Two immediate corollaries follow. First, a finite‑VC‑dimension class has finite bracketing numbers: for any ε > 0 one can build a collection of ε‑brackets (L,U) with L ⊂ A ⊂ U and μ(U \ L) ≤ ε that covers 𝔽. The partition π from the main theorem yields such brackets directly, showing N_{