If (X,X ) is a measure space and F◦ ⊂ X is a field generated either by a countable class of sets or by a Vapnik-Červonenkis class, then if μ is purely finitely additive, there exist uncountably many μ′ agreeing with μ on F◦ and having |μ(A) − μ′(A)| = 1 for uncountably many A. If μ is also non-atomic, then for any r ∈ (0, 1], |μ(Ar)− μ(Ar)| = r for uncountably many Ar. Al-Najjar’s [1] unlearnab...

Let (Ω,Σ, μ) be a purely non-atomic measure space, and let 1 < p < ∞. If L(Ω,Σ, μ) is isomorphic, as a Banach space, to L(Ω,Σ, μ) for some purely atomic measure space (Ω,Σ, μ), then there is a measurable partition Ω = Ω1 ∪Ω2 such that (Ω1,Σ ∩ Ω1, μ|Σ∩Ω1) is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable σ ⊆ Ω2. In particular, L(Ω,Σ, μ) is isomorphic to l.

We give a purely combinatorial characterization of complete Stanley-Reisner rings having a principally generated (equivalently, finitely generated) Cartier algebra.

Let Σ be a closed orientable surface, and write Map(Σ) for its mapping class group — the group of self-homeomorphisms up to homotopy. The Nielsen-Thurston classification partitions the non-trivial elements of Map(Σ) into finite order, reducible, and pseudoanosov, the last being the “generic” case. A subgroup of Map(Σ) is purely pseudoanosov if every non-trivial element is pseudoanosov. Such a s...

It is shown that the core of a non-atomic glove-market game which is defined as the minimum of finitely many non-atomic probability measures is a von Neumann Morgenstern stable set. This result is used to characterize some stable sets of large games which have a decreasing returns to scale property. We also study exact non-atomic glove-market games. In particular we show that in a glove-market ...