How to make Davies' theorem visible

We prove that for an arbitrary measurable set A subset of R-2 and a sigma - finite Borel measure mu on the plane, there is a Borel set of lines L such that for each point in A, the set of directions of those lines from L conta ining the point is a residual set, and, moreover, mu (A) = mu({U l : l is a n element of L}). We show how this result may be used to characterise the s ets of the plane from which an invisible set is visible. We also characteri se the rectifiable sets C-1, C-2 for which there is a set which is visible from C-1 and invisible from C-2.