Keisler Measures and Generically Stable Random Types
Abstract
We introduce the notions of rgs and irgs for Keisler measures, motivated by the study of generically stable random types and their associated Morley sequences. We obtain characterizations of these notions in terms of averages of classical first-order formulas over suitable probabilistic partitions (Theorems 3.2 and 3.3). We compare these notions with fim, fam, and self-averaging, and show that for types the notions fim, irgs, and rgs coincide. We prove that if a definable global Keisler measureμ is not dependent, then every Morley sequence in its canonical random extension rμ is divergent (Theorem 4.7). Consequently, every irgs measure is dependent (Corollary 4.8) and hence symmetric (Corollary 4.13). Furthermore, we show that for irgs measures the model-theoretic instability events Oφ, Iφ, and Lφ have ℙμ-measure zero (Theorem 5.4). In view of the equivalence of fim and irgs subsequently established in [6], this provides an alternative proof of the corresponding result for fim measures in [9].
Related
- Generically stable Keisler measures
- Radon measures in the Berkovich space and generically stable measures in ACVF
- Stable invariant measures in linear dynamics
- Wasserstein bounds for invariant measures in the Smoluchowski-Kramers approximation with stable noise
- Periodic measures for time-inhomogeneous stochastic differential equations driven by multiplicative α-stable processes: Existence, uniqueness and numerical approximation
BibTeX
@misc{khanaki2026keisler,
title = {{Keisler Measures and Generically Stable Random Types}},
author = {Karim Khanaki},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.15870},
eprint = {2605.15870},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.15870},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Keisler Measures and Generically Stable Random Types
AU - Karim Khanaki
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2605.15870
UR - https://arxiv.org/abs/2605.15870
AB - We introduce the notions of rgs and irgs for Keisler measures, motivated by the study of generically stable random types and their associated Morley sequences. We obtain characterizations of these notions in terms of averages of classical first-order formulas over suitable probabilistic partitions (Theorems 3.2 and 3.3). We compare these notions with fim, fam, and self-averaging, and show that for types the notions fim, irgs, and rgs coincide. We prove that if a definable global Keisler measureμ is not dependent, then every Morley sequence in its canonical random extension r_μ is divergent (Theorem 4.7). Consequently, every irgs measure is dependent (Corollary 4.8) and hence symmetric (Corollary 4.13). Furthermore, we show that for irgs measures the model-theoretic instability events O^φ, I^φ, and L^φ have ℙ_μ-measure zero (Theorem 5.4). In view of the equivalence of fim and irgs subsequently established in [6], this provides an alternative proof of the corresponding result for fim measures in [9].
ER -