A comparison of symplectic homogenization and Calabi quasi-states

A quasi-integral on a locally compact space is a certain kind of (not necessarily linear) functional on the space of continuous functions with compact support of that space. We compare two quasi-integrals on an open neighborhood of the zero section of the cotangent bundle of a circle. One comes from Viterbo's symplectic homogenization, the other from the Calabi quasi-state due to Entov and Polterovich. We provide an axiomatic description of the two functionals and a necessary and sufficient condition for them to equal. We also give a link to asymptotic Hofer geometry on T* S ¹. Proofs are based on the theory of quasi-integrals and topological measures. Finally, we give an elementary proof that a quasi-integral on a surface is symplectic.