Projective Functions

ABSTRACT

We develop an analytical calculus for projective functions on Polish spaces. In ZFC, we relate inverse‐image projective measurability to the classical graph‐based notion and establish closure under composition, finite algebraic operations, and optimization over arbitrary projective sections. Under universal measurability of projective sets, projective functions are universally measurable; under projective uniformization, projective selections and ‐optimal selectors are available. Assuming both properties, in particular, under the set‐theoretical axiom of Projective Determinacy, we establish the projective complexity of the map , stability under integration against projectively measurable stochastic kernels, and projective measurability of the multiple‐priors conditional support. The present paper supplies the analytical infrastructure for applications to model uncertainty in mathematical finance and economics; the corresponding robust utility maximization and no‐arbitrage theorems are proved elsewhere.