Sets which are not tube null and intersection properties of random measures

We show that in R-d there are purely unrectifiable sets of Hausdorff (and even box counting) dimension d - 1 which are not tube null, settling a question of Carbery, Soria and Vargas, and improving a number of results by the same authors and by Carbery. Our method extends also to 'convex tube null sets', establishing a contrast with a theorem of Alberti, Csornyei and Preiss on Lipschitz-null sets. The sets we construct are random, and the proofs depend on intersection properties of certain random fractal measures with curves.