A computational aspect of the Lebesgue differentiation theorem

Abstract

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.

Schnorr randomness and the Lebesgue differentiation theorem

“…In [14] it was shown that, for each f ∈ L 1 ([0, 1] d ) which is L 1 -computable in the sense of Definition 2.6 below, the equation (2) holds for all x ∈ [0, 1] d which are random in the sense of Martin-Löf. At the end of [14], the question of the converse was posed.…”

“…At the end of [14], the question of the converse was posed. The purpose of the present paper is to answer this question by sharpening the results of [14]. Roughly speaking, our main result is as follows:…”

“…In [14] it was shown that if x is Martin-Löf random, the Lebesgue Differentiation Theorem holds at x for all L 1 -computable functions. We now prove, in Section 3 below, that the same result holds if x is Schnorr random.…”

Lowness, Randomness, and Computable Analysis

“…Here L 1 -computability means in essence that the function can be effectively approximated by step functions, where the distance is measured in the usual L 1 -norm. About ten years later, Pathak [44] showed that ML-randomness of z suffices for the existence of the limit in the Lebesgue differentiation theorem when the given function f is L 1 -computable. This works even when f is defined on r0, 1s n for some n ą 1.…”

“…Computable randomness says that no effective betting strategy (martingale) succeeds on Z, Schnorr randomness that no such strategy succeeds quickly (see [11,40] for background). Pathak [44], followed by Pathak, Rojas and Simpson [45] characterised Schnorr randomness: Z is Schnorr random iff an effective version of the Lebesgue differentiation theorem holds at the real z P r0, 1s with binary expansion Z. Brattka, Miller and Nies [6] showed that Z is computably random if and only if every nondecreasing computable function is differentiable at z. See Section 4 for detail.…”