  We introduce a tower of localizing subcategories in Voevodsky's big (closed
under infinite coproducts) triangulated category of motives. We show that the
tower induces an interesting finite filtration on the motivic cohomology groups
of smooth schemes over a perfect field. With rational coefficients, this finite
filtration satisfies several of the properties of the still conjectural
Bloch-Beilinson-Murre filtration.
