<p>We consider the space of functions almost in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> and endow it with the topology of asymptotic <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb{R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb{R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> setting, revealing fundamental differences from standard <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> spaces.</p>

On $F$-spaces of almost-Lebesgue functions

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2025年8月1日

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N. J. Alves

摘要

We consider the space of functions almost in $L_p$ $L_{p}$ and endow it with the topology of asymptotic $L_p$ $L_{p}$ -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 $\mathbb{R}^d$ $R^{d}$ 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 $\mathbb{R}^d$ $R^{d}$ setting, revealing fundamental differences from standard $L_p$ $L_{p}$ spaces.

On \(F\)-spaces of almost-Lebesgue functions

摘要