  This paper introduces a new method to tackle the issue of the almost sure
convergence of stochastic approximation algorithms defined from a differential
inclusion. Under the assumption of slowly decaying step-sizes, we establish
that the set of essential accumulation points of the iterates belongs to the
Birkhoff center associated with the differential inclusion. Unlike previous
works, our results do not rely on the notion of asymptotic pseudotrajectories
introduced by Bena\"im--Hofbauer--Sorin, which is the predominant technique to
address the convergence problem. They follow as a consequence of Young's
superposition principle for closed measures. This perspective bridges the gap
between Young's principle and the notion of invariant measure of set-valued
dynamical systems introduced by Faure and Roth. Also, the proposed method
allows to obtain sufficient conditions under which the velocities locally
compensate around any essential accumulation point.
