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In this paper we introduce a new hierarchy of large cardinals between I3 and I2, the iterability hierarchy, and we prove that every step of it strongly implies the ones below. [ABSTRACT FROM AUTHOR]

Work in the measure algebra of the Lebesgue measure on N2: for comeagre many [A] the set of points x such that the density of x in A is not defined is Σ30‐complete; for some compact K the set of points x such that the density of x in K exists and it is different from 0 or 1 is Π30‐complete; the set of all [K] with K compact is Π30‐complete. There is a set (which can be taken to be open or closed) in R such that the density of any point is either 0 or 1, or else undefined. Conversely, if a subset of Rn is such that the density exists at every point, then the value 1/2 is always attained on comeagre many points of the measurable frontier. On the route to these results we show that the Cantor space can be embedded in a measured Polish space in a measure‐preserving fashion. [ABSTRACT FROM AUTHOR]

Abstract: Given an equivalence class in the measure algebra of the Cantor space, let be the set of points having density in . Sets of the form are called -regular. We establish several results about -regular sets. Among these, we show that -regular sets can have any complexity within (), that is for any subset of the Cantor space there is a -regular set that has the same topological complexity of . Nevertheless, the generic -regular set is -complete, meaning that the classes such that is -complete form a comeager subset of the measure algebra. We prove that this set is also dense in the sense of forcing, as -regular sets with empty interior turn out to be -complete. Finally we show that the generic does not contain a set, i.e., a set which is in . [Copyright &y& Elsevier]

Assuming AD + DC(ℝ), we characterize the self-dual boldface pointclasses which are strictly larger (in terms of cardinality) than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of Wadge rank $\leq \omega_{1}^{\xi}$, and those of Wadge rank $< \omega_{1}^{\xi}$ when ξ is limit. [ABSTRACT FROM AUTHOR]

Abstract: We show that the semi-linear ordering principle for continuous functions implies the determinacy of all Wadge and Lipschitz games. [Copyright &y& Elsevier]

We prove that the determinacy of all Lipschitz games, the determinacy of all Wadge games, and the semi-linear ordering principle for Lipschitz maps are all equivalent. [Copyright &y& Elsevier]