On theories of random variables

We study theories of spaces of random variables: first, we consider random variables with values in the interval $[0,1]$, then with values in an arbitrary metric structure, generalising Keisler's randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction: i) The randomisation of a stable structure is stable. ii) The randomisation of a simple unstable structure is not simple. We also prove that in the randomised structure, every type is a Lascar type

Stability and stable groups in continuous logic

We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity

The linear isometry group of the Gurarij space is universal

We give a construction of the Gurarij space, analogous to Katetov's construction of the Urysohn space. The adaptation of Katetov's technique uses a generalisation of the Arens-Eells enveloping space to metric space with a distinguished normed subspace. This allows us to give a positive answer to a question of Uspenskij, whether the linear isometry group of the Gurarij space is a universal Polish group

Modular functionals and perturbations of Nakano spaces

We settle several questions regarding the model theory of Nakano spaces left open by the PhD thesis of Pedro Poitevin \cite{Poitevin:PhD}. We start by studying isometric Banach lattice embeddings of Nakano spaces, showing that in dimension two and above such embeddings have a particularly simple and rigid form. We use this to show show that in the Banach lattice language the modular functional is definable and that complete theories of atomless Nakano spaces are model complete. We also show that up to arbitrarily small perturbations of the exponent Nakano spaces are $\aleph_0$-categorical and $\aleph_0$-stable. In particular they are stable

On Roeckle-precompact Polish group which cannot act transitively on a complete metric space

We study when a continuous isometric action of a Polish group on a complete metric space is, or can be, transitive. Our main results consist of showing that certain Polish groups, namely $\mathrm{Aut}^*(\mu)$ and $\mathrm{Homeo}^+[0,1]$, such an action can never be transitive (unless the space acted upon is a singleton). We also point out "circumstantial evidence" that this pathology could be related to that of Polish groups which are not closed permutation groups and yet have discrete uniform distance, and give a general characterisation of continuous isometric action of a Roeckle-precompact Polish group on a complete metric space is transitive. It follows that the morphism from a Roeckle-precompact Polish group to its Bohr compactification is surjective

Definability of groups in ℵ0\aleph_0-stable metric structures

We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to prove the theorem in case the metric is invariant under the group action; and \item Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones. \end{enumerate

Continuous and Random Vapnik-Chervonenkis Classes

We show that if $T$ is a dependent theory then so is its Keisler randomisation $T^R$. In order to do this we generalise the notion of a Vapnik-Chervonenkis class to families of $[0,1]$-valued functions (a \emph{continuous} Vapnik-Chervonenkis class), and we characterise families of functions having this property via the growth rate of the mean width of an associated family of convex compacts

On uniform canonical bases in LpL_p lattices and other metric structures

We discuss the notion of \emph{uniform canonical bases}, both in an abstract manner and specifically for the theory of atomless $L_p$ lattices. We also discuss the connection between the definability of the set of uniform canonical bases and the existence of the theory of beautiful pairs (i.e., with the finite cover property), and prove in particular that the set of uniform canonical bases is definable in algebraically closed metric valued fields