  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 \textit{weakly pspace-random} points, a new variant of polynomial
space randomness. We show that the Lebesgue differentiation theorem holds for
every weakly pspace-random point.
