Projective functions

TL;DR

This paper investigates projective functions, demonstrating their generalization of semianalytic functions, their stability under various operations, and their applications in model uncertainty, with results contingent on the axiom of Projective Determinacy.

Contribution

It establishes the stability properties of projective functions under multiple operations and explores their measurable selection and integration properties under Projective Determinacy.

Findings

Projective functions generalize lower and upper-semianalytic functions.

They are stable under composition, difference, sum, product, and other operations.

Results depend on the axiom of Projective Determinacy.

Abstract

We study projective functions. We prove that projective functions generalise lower and upper-semianalytic ones while being stable by composition and difference. We show that the class of projective functions is closed under sums, differences, products, finite suprema and infima, sections and compositions. Assuming the set-theoretical axiom of Projective Determinacy, we also prove measurable selection results, stability under integration, and the existence of $\epsilon$-optimal selectors. Finally, we illustrate how these results are important in the context of model uncertainty.