On $\mathrm{F}$-spaces of almost-Lebesgue functions

TL;DR

This work introduces the almost-Lp spaces $\Lambda_p(X)$ equipped with the F-norm $\|\cdot\|_{\alpha_p}$, establishing that they form a complete, metrizable F-space whose topology matches asymptotic $L_p$ convergence. It proves metrizability/completeness and, on finite measure spaces, that $\Lambda_p(X)$ coincides with $L_0(X)$ under convergence in measure, while for $\mathbb{R}^d$ it reveals substantial structural differences from standard $L_p$ spaces. The authors develop dominated and Vitali-type convergence theorems in this framework, study approximation and separability (including density of $C_c^\infty$ and separability of $\Lambda_p(\mathbb{R}^d)$), and show that $\Lambda_p(\mathbb{R}^d)$ is neither locally bounded nor locally convex with a trivial dual, highlighting new phenomena in F-spaces of measurable functions.

Abstract

We consider the space of functions almost in $L_p$ and endow it with the topology of asymptotic $L_p$-convergence. This yields a completely metrizable topological vector space which, on finite measure spaces, coincides with the space of measurable functions equipped with the topology of (local) convergence in measure. We investigate analogs of classical results such as dominated convergence and Vitali convergence theorems. For $\mathbb{R}^d$ as the underlying measure space, we establish results on approximation by smooth functions and separability. Further aspects, including local boundedness, local convexity, and duality are examined in the $\mathbb{R}^d$ setting, revealing fundamental differences from standard $L_p$ spaces.

Theorems & Definitions (51)

• Theorem 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Example 2.4
• Proposition 2.5
• proof