Shape optimization problems on metric measure spaces

We consider shape optimization problems of the formmin{J(Ω):Ω⊂X,m(Ω)⩽c}, where X is a metric measure space and J is a suitable shape functional. We adapt the notions of γ-convergence and weak-γ convergence to this new general abstract setting to prove the existence of an optimal domain. Several examples are pointed out and discussed.