  In the first part of the paper, we define an approximated Brunn-Minkowski
inequality which generalizes the classical one for length spaces. Our new
definition based only on distance properties allows us also to deal with
discrete spaces. Then we show the stability of our new inequality under a
convergence of metric measure spaces. This result gives as a corollary the
stability of the classical Brunn-Minkowski inequality for geodesic spaces. The
proof of this stability was done for different inequalities (curvature
dimension inequality, metric contraction property) but as far as we know not
for the Brunn-Minkowski one. In the second part of the paper, we show that
every metric measure space satisfying classical Brunn-Minkowski inequality can
be approximated by discrete spaces with some approximated Brunn-Minkowski
inequalities.
