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 ε/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.

Measuring associations between variables is one of the central goals of data analysis. Arguably, the three most popular classical measures of association are Pearson's correlation coefficient, Spearman's ρ, and Kendall's τ. Although these coefficients are powerful for detecting monotonic associations, a practical problem is that they are not effective for detecting associations that are not monotonic.

We present a universally consistent learning rule whose expected error is monotone non-increasing with the sample size under every data distribution. The question of existence of such rules was brought up in 1996 by Devroye, Gy\"orfi and Lugosi (who called them "smart"). Our rule is fully deterministic, a data-dependent partitioning rule constructed in an arbitrary domain (a standard Borel space) using a cyclic order. The central idea is to only partition at each step those cyclic intervals that exhibit a sufficient empirical diversity of labels, thus avoiding a region where the error function is convex.

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.