In this paper we study the existence of Lebesgue densities of one-dimensional Lévy processes. Equivalently, we show the absolute continuity of the distributions of one-dimensional Lévy processes. Compared with the previous literature, we consider Lévy processes with Lévy symbols of a logarithmic behavior at ∞.

We study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable functions. The corresponding randomness notions are slightly stronger than Martin–Löf (ML) randomness.

We establish several equivalences. Given a ML-random real z, the additional randomness strengths needed for the following are equivalent.

1. (1) all effectively closed classes containing z have density 1 at z.

2. (2) all nondecreasing functions with uniformly left-c.e. increments are differentiable at z.

3. (3) z is a Lebesgue point of each lower semicomputable integrable function.

We also consider convergence of left-c.e. martingales, and convergence in the sense of Birkhoff’s pointwise ergodic theorem. Lastly, we study randomness notions related to density of ${\rm{\Pi }}_n^0$ and ${\rm{\Sigma }}_1^1$ classes at a real.