  We study the asymptotic behavior of volume forms on a degenerating family of
compact complex manifolds. Under rather general conditions, we prove that the
volume forms converge in a natural sense to a Lebesgue-type measure on a
certain simplicial complex. In particular, this provides a measure-theoretic
version of a conjecture by Kontsevich--Soibelman and Gross--Wilson, bearing on
maximal degenerations of Calabi--Yau manifolds.
