  Let $k$ be a field admitting resolution of singularities. We lift a number of
motivic measures, such as the Gillet-Soul\'e measure and the compactly
supported $\mathbb{A}^1$-Euler characteristic, to derived motivic measures in
the sense of Campbell-Wolfson-Zakharevich, answering various questions in the
literature. We do so by generalizing the construction of the Gillet-Soul\'e
weight complex to show that it is well-defined up to a certain notion of weak
equivalence in the category of simplicial smooth projective varieties. For a
$k$-variety $X$, the collection of all Gillet-Soul\'e weight complexes of $X$
form a 'weakly constant' pro-object of simplicial varieties, and under mild
assumptions, the $K$-theory of a Waldhausen category is equivalent to the
$K$-theory of its weakly constant pro-objects. This leads us to a new proof of
the existence of the Gillet-Soul\'e weight filtration, along with the weight
filtration on both the stable and unstable homotopy type of a variety over $k$.
We show these constructions provide the aforementioned derived motivic
measures, or maps of spectra, out of $K(Var_k)$, the Zakharevich $K$-theory of
varieties.
