Abstract. We exhibit a close correspondence between L 1 -computable functions and Schnorr tests. Using this correspondence, we prove that a point x ∈ [0, 1] d is Schnorr random if and only if the Lebesgue Differentiation Theorem holds at x for all L 1 -computable functions f ∈ L 1 ([0, 1] d ).

Given an L 1-computable function, f , we identify a canonical representative of the equivalence class of f , where f and g are equivalent if and only if |f − g| is zero. Using this representative, we prove a modified version of the Lebesgue Differentiation Theorem. Our theorem is stated in terms of Martin-Löf random points in Euclidean space.

Given an L 1-computable function, f , we identify a canonical representative of the equivalence class of f , where f and g are equivalent if and only if |f − g| is zero. Using this representative, we prove a modified version of the Lebesgue Differentiation Theorem. Our theorem is stated in terms of Martin-Löf random points in Euclidean space.