  This article is the continuation of [LS12]. We use categories of matrix
factorizations to define a morphism of rings (= a Landau-Ginzburg motivic
measure) from the (motivic) Grothendieck ring of varieties over $\mathbb{A}^1$
to the Grothendieck ring of saturated dg categories (with relations coming from
semi-orthogonal decompositions into admissible subcategories). Our
Landau-Ginzburg motivic measure is the analog for matrix factorizations of the
motivic measure in [BLL04] whose definition involved bounded derived categories
of coherent sheaves. On the way we prove smoothness and a Thom-Sebastiani
theorem for enhancements of categories of matrix factorizations.
