In this lecture, we first introduce a new notion of generalized super-weak Funk functions on a Finsler (or spray) space. Then we show that there is no non-trivial generalized super-weak Funk function on a complete spray space generalizing a result previously known only in the case of weak Funk functions on compact spray spaces. As its application, we prove that every complete Finsler space with ${\bf S}\geq\lambda\tau^2$ for some positive constant $\lambda$ must be Riemannian where ${\bf S}$ and $\tau$ are the S-curvature and the distortion respectively. We also give {\em new} or {\em simple} proofs of the Shen-Sun's and Chen-Shen's global rigidity theorems.
