Weak Convergence of Robust Functions on Topological Groups

This paper introduces weak variants of level convergence (L-convergence) and epigraph convergence (E-convergence) for nets of level functions on general topological spaces, extending the classical metric and real-valued frameworks to ordered codomains and generalized minima. We show that L-convergence implies E-convergence and that the two notions coincide when the limit function is level-continuous, mirroring the relationship between strong and weak variational convergence. In Hausdorff topological groups, we define robust level functions and prove that every level function can be approximated by robust ones via convolution-type operations, enabling perturbation-resilient modeling. These results both generalize and connect to Γ-convergence: they recover the classical metric, lower semicontinuous case, and extend the scope for optimization on Lie groups, fuzzy systems, and mechanics in non-Euclidean spaces. An explicit nonmetrizable example demonstrates the relevance of our theory beyond the reach of Γ-convergence.