Abstract: 

Continuous logic generalizes first-order logic to take real truth values, and Keisler measures similarly generalize types. We combine these two generalizations to study measures in continuous logic, focusing on generically stable measures, which are particularly well-behaved. In NIP theories, generically stable measures have several equivalent definitions. We translate these definitions to continuous logic, and show that they remain equivalent in this context. This requires developing real-valued versions of combinatorial theorems about classes of finite VC-dimension, and extrapolating definitions like smoothness and weak orthogonality of measures to continuous logic. We then use generically stable measures to derive regularity lemmas for NIP and distal metric structures.

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