Schnorr randomness and the Lebesgue differentiation theorem (Q2862195)

The authors show that a point \(x\in [0,1]^d\) is Schnorr random if and only if the Lebesgue differentiation theorem holds at~\(x\) for every \(L_1\)-computable function. This result is part of a developing project relating notions of algorithmic randomness to effective versions of almost-everywhere theorems of analysis and ergodic theory. Previous results related both Schnorr and Martin-Löf randomness to Birkhoff's ergodic theorem; More recently, \textit{V. Brattka}, \textit{J. S. Miller} and \textit{A. Nies} [``Randomness and differentiability'', Trans. Am. Math. Soc. (to appear)] related both computable randomness and ML-randomness to differentiability of monotone functions and functions of bounded variation.