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Borel selection of dominating hyperplanes

Eugenio Clerico

Abstract

We study a natural measurable selection problem for which the standard uniformisation theorems do not seem to apply directly, yet a Borel selector exists. More precisely, we consider families of finite dimensional functions that admit pointwise domination by affine functionals and ask whether such dominating functionals can be chosen in a Borel measurable way. We prove that this is indeed possible under semi-analytic regularity assumptions. The proof combines a one-dimensional Borel insertion result between an upper and a lower semi-analytic functions, derived from Lusin's separation theorem, with an induction on the dimension. As an application, we obtain Borel measurable selections of subgradients for parameter-dependent finite-dimensional convex functions, outside the scope of the standard normal integral framework.

Borel selection of dominating hyperplanes

Abstract

We study a natural measurable selection problem for which the standard uniformisation theorems do not seem to apply directly, yet a Borel selector exists. More precisely, we consider families of finite dimensional functions that admit pointwise domination by affine functionals and ask whether such dominating functionals can be chosen in a Borel measurable way. We prove that this is indeed possible under semi-analytic regularity assumptions. The proof combines a one-dimensional Borel insertion result between an upper and a lower semi-analytic functions, derived from Lusin's separation theorem, with an induction on the dimension. As an application, we obtain Borel measurable selections of subgradients for parameter-dependent finite-dimensional convex functions, outside the scope of the standard normal integral framework.
Paper Structure (8 sections, 6 theorems, 23 equations)

This paper contains 8 sections, 6 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analytic sandwich lemma
  4. Hyperplane selection
  5. Linear functional selection
  6. Borel selection of subgradients
  7. Discussion
  8. Acknowledgments.

Key Result

Lemma 1

Let $\mathcal{X}$ be a standard Borel space, let $u:\mathcal{X}\to\mathbb R$ be usa and $l:\mathcal{X}\to\mathbb R$ be lsa. If $u\leq l$, pointwise on $\mathcal{X}$, then there exists a Borel function $f:\mathcal{X}\to\mathbb R$ such that $u\leq f\leq l$, pointwise on $\mathcal{X}$.

Theorems & Definitions (12)

  • Lemma 1: Analytic sandwich
  • proof
  • Lemma 2
  • proof
  • Theorem 1: Hyperplane selection
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • ...and 2 more