To the question of realizability of the Crofton function on sets in \(\mathbb{R}^ 2\) (Q1322648)

In this short note the following theorem is proven. Let \(S\) be a subset of the plane with the property that it intersects every line in a set of size continuum. Suppose that \(F\) is function which assigns to each line a member of the set \(\{2,3,\dots, \infty\}\). Then there exists a subset \(E\) of \(S\) such that \(E\) meets \(l\) in exactly \(F(l)\) points for every line \(l\). This is a generalization of Mazurkiewicz's result that there exists a subset of the plane which meets every line in exactly two points. Related results in the literature are discussed.