  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 construct Chern--Schwartz--MacPherson classes of
proalgebraic varieties, by introducing the notion of ``proconstructible
functions " and "$\chi$-stable proconstructible functions" and using the
Fulton-MacPherson's Bivariant Theory. As a "motivic" version of a $\chi$-stable
proconstructible function, $\Ga$-stable constructible functions are introduced.
This construction naturally generalizes the so-called motivic measure and
motivic integration. For the Nash arc space $\Cal L(X)$ of an algebraic variety
$X$, the proconstructible set is equivalent to the so-called cylinder set or
constructible set in the arc space.

Algebraic Geometry;Algebraic Topology