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

TL;DR

This paper studies a new topological space of functions nearly in $L_p$, exploring its properties and classical convergence theorems, with special focus on $ ext{R}^d$, revealing fundamental differences from standard $L_p$ spaces.

Contribution

It introduces a novel space of almost-$L_p$ functions with an asymptotic convergence topology and investigates its properties, including approximation, separability, and duality, highlighting key differences from classical $L_p$ spaces.

Findings

The space is completely metrizable and coincides with local convergence in measure on finite measure spaces.

Classical theorems like dominated and Vitali convergence are extended to this space.

Results on approximation by smooth functions and separability are established for $ ext{R}^d$.

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.