  We introduce a notion of integration on the category of proper birational
maps to a given variety $X$, with value in an associated Chow group.
Applications include new birational invariants; comparison results for Chern
classes and numbers of nonsingular birational varieties; `stringy' Chern
classes of singular varieties; and a zeta function specializing to the
topological zeta function.
  In its simplest manifestation, the integral gives a new expression for
Chern-Schwartz-MacPherson classes of possibly singular varieties, placing them
into a context in which a `change-of-variable' formula holds. v2: References
added, and overly optimistic claims concerning non log-terminal singularities
expunged.
