Categorical directional density and some related results (Q1346462)

The main result of the paper is the following Theorem: If \(B\) is a subset of the plane having the Baire property, then for each point belonging to \(B\) except a set of the first category (as a plane set) this point is a point of categorical density in all directions except a first category set of directions. The authors claim that they use the definition of linear category density according to the reviewer [Real Anal. Exch. 10(1984/85), 241-265 (1985; Zbl 0593.26008)]. However, their definition is not equivalent. Roughly speaking, they use the convergence almost everywhere instead of the convergence in measure.