It is shown that there exist a continuous function f and a regulated function g defined on the interval [0, 1] such that g vanishes everywhere except for a countable set, and the K-integral of f with respect to g does not exist. The problem was motivated by extensions of evolution variational inequalities to the space of regulated functions.

If a function f has finite Henstock integral on the boundary of the unit disk of R 2 then its Poisson integral exists for |z| < 1 and is o((1 − |z|) −1) as |z| → 1 −. It is shown that this is the best possible uniform pointwise estimate. For an L 1 measure the best estimate is O((1 − |z|) −1). In this paper we consider estimates of Poisson integrals on the unit circle with respect to Alexiewicz...