  Making use of the recent theory of noncommutative motives, we construct a new
motivic measure, which we call the Tits' motivic measure. As a first
application, we prove that two Severi-Brauer varieties (or more generally
twisted Grassmannian varieties), associated to central simple algebras of
period 2, have the same Grothendieck class if and only if they are isomorphic.
As a second application, we show that if two Severi-Brauer varieties,
associated to central simple algebras of period 2, 3, 4, 5 or 6, have the same
Grothendieck class, then they are necessarily birational. As a third
application, we prove that two quadric hypersurfaces (or more generally
involution varieties), associated to quadratic forms of degree 6, have the same
Grothendieck class if and only if they are isomorphic. This latter result also
holds for products of such quadrics. Finally, as a fourth application, we show
in certain cases that two products of conics have the same Grothendieck class
if and only if they are isomorphic; this refines a result of Kollar.
