In response to a 1997 problem of M. Vidyasagar, we state a necessary and sufficient condition for distribution-free PAC learnability of a concept class C under the family of all non-atomic (diffuse) measures on the domain Ω. Clearly, finiteness of the classical Vapnik–Chervonenkis dimension of C is a sufficient, but no longer necessary, condition. Besides, learnability of C under non-atomic mea...

The findings reported in this paper aim to garner the interest of both model theorists and operator algebraists alike. Using a novel blend theoretic algebraic methods, we show that family II 1 factors elementarily equivalent hyperfinite factor R all admit embeddings into U with factorial relative commutant. This answers long standing question Popa for an uncountable factors. We introduce notion...

The relative realizability toposes that Awodey, Birkedal and Scott introduced in [1] satisfy a universal property that involves regular functors to other categories. We use this universal property to define what relative realizability categories are, when based on other categories than of the topos of sets. This paper explains the property and gives a construction for relative realizability cat...

as a straightforward generalization of (1). To understand the convergence properties of αn it is a natural question to investigate the quantities (2) sup C∈C |αn(C)| and sup f∈F |αn(f)|, respectively. In the simple case of ξ1, ξ2, . . . being “ordinary” real random variables and C being the class of subintervals of R, a result of this type is the well-known Glivenko–Cantelli theorem. In the gen...