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BIRS Workshop Lecture Videos

Schnorr randomness for noncomputable measures Rute, Jason


Schnorr randomness is a randomness notion based on Brouwer's concept of a "constructive null set." Recently, Schnorr randomness has been closely associated with a number of theorems in computable analysis, including the Lebesgue differentiation theorem, the ergodic theorem (for ergodic measures), and Carleson's theorem. Nonetheless, the theory of Schnorr randomness is not nearly as developed as that of Martin-Löf randomness. In particular, there is no established notion of Schnorr randomness with respect to noncomputable measures. Such a notion would be essential to applying Schnorr randomness to disintegration theorems such as de Finetti's theorem or the ergodic decomposition theorem. In this talk I will present a novel definition of Schnorr randomness for noncomputable measures. Say that $x_0$ is \emph{Schnorr random with respect to} a (possibly noncomputable) measure $\mu_0$ if $t(x_0,\mu_0) < \infty$ for all lower semicomputable functions $t(x,\mu)$ such that $\mu \mapsto \int t(x,\mu) d\mu$ is finite and computable. I will show that this definition satisfies the basic properties that one would expect such a notion to have. I will also present various applications of this definition, and discuss how it fits into a larger research program.

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