Gromov–Hausdorff variational principles and measure stability *

Variational formulae for the Gromov–Hausdorff distances of compact metric spaces and their continuous maps are obtained. Also, a notion of µ-topological GH-stability on metric-measure spaces with probability µ is given. We prove that every expansive map with the weak µ -shadowing property (Lee and Rojas 2022 Russ. J. Nonlinear Dyn. 18 297–307) is µ -topologically GH-stable.