Let μ be a Borel charge (i. e., a real Borel measure) on ℝ, and let P (y) (t)=y/π(y 2+t 2),y > 0, t ∈ ℝ, denote the Poisson kernel. Bourgain proved that for a nonnegative μ and for many points t ∈ ℝ, the variation of the function y{mapping}(μ*P (y)) on (0, 1] is finite. This is true, in particular, for so-called B-points x introduced in a previous author's paper, In the present paper, we give new descriptions of B-points which are adjusted to some applications of this notion. Bibliography: 5 titles. © 2012 Springer Science+Business Media, Inc.