In the rst part of the paper, we give a new de nition of Brunn-
Minkowski inequality in metric measure space. Then we show the stability
of Brunn-Minkowski inequality under a convergence of metric measure
space. This result gives as a corollary the stability of the classical Brunn-
Minkowski inequality for geodesic spaces.
In the second part, we show that every metric measure space satisfying
Brunn-Minkowski inequality can be approximated by discrete space with
some approximated Brunn-Minkowski inequalities.
