Null Measurability at the Symmetrization Interface in VC Learning

TL;DR

This paper demonstrates that weaker measurability conditions than Borel are sufficient for symmetrization in VC learning, broadening the theoretical foundations of PAC learnability.

Contribution

It shows that the standard Borel measurability requirement can be relaxed to analytic measurability, with stability under natural concept class operations.

Findings

The bad event is analytic and measurable in the completion of every finite Borel measure.

Constructed a concept class with a null-measurable but non-Borel bad event.

Proved closure properties under patching, interpolation, and fiber-product amalgamation.

Abstract

Recent work revisiting measurability in the fundamental theorem of statistical learning imposes Borel measurability of ghost-gap suprema. We show that, at the one-sided ghost-gap interface actually used by the standard symmetrization proof, this requirement is stronger than necessary. For any Borel-parameterized concept class on a Polish domain, the bad event "there exists a hypothesis whose ghost empirical error exceeds its training empirical error by at least {\epsilon}/2" is analytic. By Choquet capacitability, it is therefore measurable in the completion of every finite Borel measure. We then construct a concept class whose bad event is null-measurable but not Borel, giving a strict separation from the Borel supremum condition. Finally, we prove closure under patching, fixed and countable interpolation, and fiber-product amalgamation, showing that the weaker regularity level is stable under natural concept-class constructors. In the realizable setting, where targets belong to the class and are measurable, these results weaken the measurability hypothesis needed by the symmetrization route from finite VC dimension to PAC learnability. The main results and the descriptive-set-theoretic infrastructure used by them are formalized in Lean 4.