Applications of descriptive set theory in mathematical analysis

We characterize various types of σ-porosity via an infinite game in terms of winning strategies. We use a modification of the game to prove and reprove some new and older in- scribing theorems for σ-ideals of σ-porous type in locally compact metric spaces. We show that there exists a closed set which is σ-(1 − ε)-symmetrically porous for every 0 < ε < 1 but which is not σ-1-symmetrically porous. Next, we prove that the realizable by an action unitary representations of a finite abelian group Γ on an infinite-dimensional complex Hilbert space H form a comeager set in Rep(Γ, H).