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On Lebesgue points and measurability with Choquet integrals

Petteri Harjulehto, Ritva Hurri-Syrjänen

Abstract

We consider Choquet integrals with respect to dyadic Hausdorff content of non-negative functions which are not necessarily Lebesgue measurable. We study the theory of Lebesgue points. The studies yield convergence results and also a density result between function spaces. We provide examples which show sharpness of the main convergence theorem. These examples give additional information about the convergence in the norm also, namely the difference of the functions in this setting and continuous functions.

On Lebesgue points and measurability with Choquet integrals

Abstract

We consider Choquet integrals with respect to dyadic Hausdorff content of non-negative functions which are not necessarily Lebesgue measurable. We study the theory of Lebesgue points. The studies yield convergence results and also a density result between function spaces. We provide examples which show sharpness of the main convergence theorem. These examples give additional information about the convergence in the norm also, namely the difference of the functions in this setting and continuous functions.

Paper Structure

This paper contains 6 sections, 18 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Hausdorff content
  3. Choquet integral
  4. Auxiliary convergence results
  5. Lebesgue points and covergence results
  6. Counter examples

Key Result

Corollary 1.1

Let $n\geq 1$. Suppose that $f:{\varmathbb{R}^n} \to [0, \infty]$ is a function such that $\int_{B} f(y) \, d\tilde{\mathcal{H}}^n_\infty (y) < \infty$ for all open balls $B$ in ${\varmathbb{R}^n}$. Then, the function $f$ is Lebesgue measurable if and only if for $\tilde{\mathcal{H}}^n_\infty$-almost every $x \in {\varmathbb{R}^n}$.

Theorems & Definitions (41)

  • Corollary 1.1
  • Lemma 3.2
  • Remark 3.3
  • Theorem 3.5
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Definition 4.3
  • Proposition 4.5
  • ...and 31 more