On the differentiation of integrals with respect to translation invariant convex density bases

Giorgi Oniani Fundamenta Mathematicae MSC: 28A15, 42B25. DOI: 10.4064/fm613-8-2018 Published online: 18 February 2019

Abstract

For a translation invariant convex density basis $B$ it is shown that its Busemann–Feller extension $B_{\mathrm {BF}}$ has properties close to $B$, namely $B_{\mathrm {BF}}$ differentiates the same class of non-negative functions as $B$, and the integral of an arbitrary non-negative function $f\in L(\mathbb {R}^n)$ at almost every point $x\in \mathbb {R}^n$ has the same type limits of indeterminacy with respect to the bases $B$ and $B_{\mathrm {BF}}$. This theorem provides a certain general principle of extending results obtained for Busemann–Feller bases to results for bases without the Busemann–Feller property. Applications of the theorem are given.

Authors

  • Giorgi OnianiDepartment of Mathematics
    Akaki Tsereteli State University
    59 Tamar Mepe St.
    Kutaisi 4600, Georgia
    e-mail

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