Continuous stable regularity.

We prove an analytic version of the stable graph regularity lemma by Malliaris and Shelah (Trans. Amer. Math. Soc. 366 (2014), no. 3, 1551–1585), which applies to stable functions f:V×W→[0,1]$f\colon V\times W\rightarrow [0,1]$. Our methods involve continuous model theory and, in particular, results on the structure of local Keisler measures for stable continuous formulas. Along the way, we develop some basic tools around ultraproducts of metric structures and linear functionals on continuous formulas, and we also describe several concrete families of examples of stable functions. [ABSTRACT FROM AUTHOR]