Definability of groups in ℵ<sub>0</sub>-stable metric structures

Abstract. We give a model-theoretic account for several results regarding sequences of random variables appearing in Berkes & Rosenthal [BR85]. In order to do this,• We study and compare three notions of convergence of types in a stable theory: logic convergence, i.e., formula by formula, metric convergence (both already well studied) and convergence of canonical bases. In particular, we characterise ℵ 0 -categorical stable theories in which the last two agree.• We characterise sequences which admit almost indiscernible sub-sequences.• We apply these tools to ARV , the theory (atomless) random variable spaces. We characterise types and notions of convergence of types as conditional distributions and weak/strong convergence thereof, and obtain, among other things, the Main Theorem of Berkes & Rosenthal.

show abstract

“….5 ([Ben10a] or[BBHU08]). Let M be a structure, X ⊆ M a closed, possibly large subset, A ⊆ M a set of parameters.…”

mentioning

confidence: 99%

Almost Indiscernible Sequences and Convergence of Canonical Bases

Yaacov

Berenstein²,

Henson

2014

J. symb. log.

View full text Add to dashboard Cite

show abstract

“…, n} (see Lemma 1.8.15 of [4]). Thus,B U n (x; δ) is a cartesian product of definable sets, whence definable by Lemma 1.10 of [2]. Thus f (B U n (x; δ)) is a closed subset of C. It follows thatf (B U n (x; δ)) = {y} and hencef −1 (y) is a clopen subset of U. Thusf (U n ) = {y}.…”

Section: The Case Of Compact Rangementioning

confidence: 93%

“…Finally, if X is a definable subset of U n , then, by ω 1saturation,f (X) is a closed subset of U. (In fact,f (X) is a definable subset of U, although we will not need this fact; see [2], Lemma 1.20.) As in classical model theory, if f : U n → U is A-definable, then, for all x ∈ U n , we havef (x) ∈ dcl(Ax).…”

Section: Notations and Backgroundmentioning

confidence: 99%

Definable functions in Urysohn’s metric space

Goldbring¹

2011

Illinois J. Math.

View full text Add to dashboard Cite

Let U denote the Urysohn sphere and consider U as a metric structure in the empty continuous signature. We prove that every definable function U n → U is either a projection function or else has relatively compact range. As a consequence, we prove that many functions natural to the study of the Urysohn sphere are not definable. We end with further topological information on the range of the definable function in case it is compact.

show abstract

“…We start with the following lemma which shows that continuous logic can be applied only to metric groups which have bi-invariant metrics. This lemma appears as a part of Proposition 3.13 in [2]. Lemma 2 Let a group (G, d) be a metric L-structure with respect to continuity moduli γ 1 and γ 2 as above.…”

Section: Theorem 1 Let G Be a Locally Compact Group With A Left-invarmentioning

confidence: 99%