Polynomial Space Randomness in Analysis

We study the interaction between polynomial space randomness and a fundamental result of analysis, the Lebesgue differentiation theorem. We generalize Ko’s framework for polynomial space computability in ${\mathbb{R}}^n$ to define $\mathit{\text{weakly}}\mathit{\text{pspace-random}}$ points, a new variant of polynomial space randomness. We show that the Lebesgue differentiation theorem holds for every weakly pspace-random point.