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Projective functions

Laurence Carassus, Massinissa Ferhoune

TL;DR

The paper defines projective functions as $\Delta_n^1(X)$-measurable on projective domains and shows they generalize lower and upper semianalytic functions. It establishes stability under key operations (including composition, sums, products, finite suprema/infima, and sections) and provides a precise equivalence between measurability and graph-projectivity. Under the axiom of Projective Determinacy ($\mathrm{PD}$), it proves measurable selection, universal measurability, and $\epsilon$-optimal selectors for projective problems, as well as the preservation of projectivity under integration with projectively measurable kernels. The results are motivated by and applied to model uncertainty in finance and economics, offering a unified, projective-measurability framework that extends prior analytic-based approaches. Overall, the work furnishes a robust mathematical foundation for dynamic decision-making under Knightian uncertainty and connects descriptive set theory with practical optimization in uncertain environments.

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 $ε$-optimal selectors. Finally, we illustrate how these results are important in the context of model uncertainty.

Projective functions

TL;DR

The paper defines projective functions as -measurable on projective domains and shows they generalize lower and upper semianalytic functions. It establishes stability under key operations (including composition, sums, products, finite suprema/infima, and sections) and provides a precise equivalence between measurability and graph-projectivity. Under the axiom of Projective Determinacy (), it proves measurable selection, universal measurability, and -optimal selectors for projective problems, as well as the preservation of projectivity under integration with projectively measurable kernels. The results are motivated by and applied to model uncertainty in finance and economics, offering a unified, projective-measurability framework that extends prior analytic-based approaches. Overall, the work furnishes a robust mathematical foundation for dynamic decision-making under Knightian uncertainty and connects descriptive set theory with practical optimization in uncertain environments.

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 -optimal selectors. Finally, we illustrate how these results are important in the context of model uncertainty.

Paper Structure

This paper contains 15 sections, 20 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Projective sets
  3. Definition of projective sets
  4. Properties of projective sets
  5. Projective functions
  6. Definition of projective functions
  7. Property of general projective functions
  8. Properties of real or vector-valued projective functions
  9. Axiom of Projective Determinacy
  10. Consequences of the (PD) axiom
  11. Properties of projective functions under the (PD) axiom
  12. Existence of $\epsilon$-optimizer for projective functions
  13. Integration of projective functions
  14. Applications to mathematical finance and economic
  15. Appendix

Key Result

Proposition 1

Let $n\geq 1,$$X$ and $Y$ be Polish spaces. Let $f : X \to Y$ be Borel measurable. (i) The class $\Sigma_n^1(X)$ is closed under countable intersections and unions. Moreover, we have that $f^{-1}(B)\in \Sigma_n^1(X)$ for all $B\in\Sigma_n^1(Y)$ and $f(A)\in\Sigma_n^1(Y)$ for all $A\in\Sigma_n^1(X)$. (vi) The class $\textbf{P}(X)$ is closed under finite unions and intersections. It is also closed u

Theorems & Definitions (31)

  • Definition 1
  • Proposition 1
  • Remark 1
  • Remark 2
  • Claim 1
  • Definition 2
  • Remark 3
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • ...and 21 more