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Relation between asymptotic $L_p$-convergence and some classical modes of convergence

Nuno J. Alves, Giorgi G. Oniani

TL;DR

The paper investigates how asymptotic $L_p$-convergence ($\alpha_p$-convergence) relates to classical convergence notions, specifically convergence in measure and convergence in weak $L_p$ spaces. It defines $\alpha_p$-convergence and proves that on finite measure spaces this notion is equivalent to convergence in measure, while also clarifying the relationship between $\alpha_p$-convergence and $\alpha_p$-Cauchy sequences. It further analyzes the interaction with weak $L_p$ spaces, providing counterexamples that show $\alpha_p$-convergence and $L_{p,\infty}$-convergence need not imply each other, though convergence in $L_{p,\infty}$ does imply $\alpha_p$-convergence in finite measure spaces. Finally, it introduces almost $L_p$ spaces $A_p$, showing nontrivial separations from $L_{p,\infty}$ in general, and establishes $L_{p,\infty}\subseteq A_p$ when the underlying measure space has finite measure.

Abstract

Asymptotic $L_p$-convergence, which resembles convergence in $L_p$, was introduced to address a question in diffusive relaxation. This note aims to compare asymptotic $L_p$-convergence with convergence in measure and in weak $L_p$ spaces. One of the results characterizes convergence in measure on finite measure spaces in terms of asymptotic $L_p$-convergence.

Relation between asymptotic $L_p$-convergence and some classical modes of convergence

TL;DR

The paper investigates how asymptotic -convergence (-convergence) relates to classical convergence notions, specifically convergence in measure and convergence in weak spaces. It defines -convergence and proves that on finite measure spaces this notion is equivalent to convergence in measure, while also clarifying the relationship between -convergence and -Cauchy sequences. It further analyzes the interaction with weak spaces, providing counterexamples that show -convergence and -convergence need not imply each other, though convergence in does imply -convergence in finite measure spaces. Finally, it introduces almost spaces , showing nontrivial separations from in general, and establishes when the underlying measure space has finite measure.

Abstract

Asymptotic -convergence, which resembles convergence in , was introduced to address a question in diffusive relaxation. This note aims to compare asymptotic -convergence with convergence in measure and in weak spaces. One of the results characterizes convergence in measure on finite measure spaces in terms of asymptotic -convergence.
Paper Structure (3 sections, 3 theorems, 20 equations)

This paper contains 3 sections, 3 theorems, 20 equations.

Table of Contents

  1. Asymptotic $L_p$-convergence vs. convergence in measure
  2. Asymptotic $L_p$-convergence vs. convergence in weak $L_p$ spaces
  3. Almost $L_p$ spaces vs. weak $L_p$ spaces

Key Result

Theorem 1.1

Let $f_n$$(n\in \mathbb{N})$ and $f$ be measurable functions. If $(f_n)$$\alpha_p$-converges to $f$, then $(f_n)$ converges to $f$ in measure. On the other hand, if $\mu(X) < \infty$ and $(f_n)$ converges to $f$ in measure, then $(f_n)$$\alpha_p$-converges to $f$.

Theorems & Definitions (10)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • proof
  • Example 2.1
  • Example 2.2
  • Example 3.1
  • Example 3.2
  • Proposition 3.3
  • proof