Michael Gromov has recently initiated what he calls ``symbolic algebraic
geometry", in which objects are proalgebraic varieties: a proalgebraic variety
is by definition the projective limit of a projective system of algebraic
varieties. In this paper we introduce characteristic classes of proalgebraic
varieties, using Grothendieck transformations of Fulton--MacPherson's Bivariant
Theory, modeled on the construction of MacPherson's Chern class transformation
of proalgebraic varieties. We show that a proalgebraic version of the
Euler--Poincar\'e characteristic with values in the Grothendieck ring is a
generalization of the so-called motivic measure.Comment: a revised version of math.AG/040723